Discovery writing doesn't require a plot, exactly. It does require a problem, undesirable situation, or confrontation to get started; all of these are under the banner of "conflict", and I +1 Galestel for describing how to find some conflict for your characters.
I won't repeat that; but I will note conflict is not necessarily a fight (physical or verbal), it is any problem the MC must deal with. It can even be an entirely mental conflict: guilt (e.g. over cheating), misunderstandings, lost love, urges they cannot control but wish they could, an important scientific problem they are struggling to solve. Anything requiring a struggle and work to overcome, even if the struggle is just in understanding.
Conflict can be external and environmental; the setting can the villain. The power has gone out. The coffee maker doesn't work. There has been an earthquake. The ship has malfunctioned and our heroes are on a strange and dangerous planet with no way home. (e.g. Tom Hanks in "Castaway", no evil person sabotaged the aircraft he was on.)
On to Discovery Writing: The trick here is to understand, up front, you will likely write many pages you intended for publication but will scrap, but that writing serves a purpose: clarifying your characters and your plot. This is how you do the work to understand, for the characters you have defined, what their story must be.
I believe if you have a few cool characters, subconsciously you have the story in you; how they interact will define what their conflict must be, and what their setting must be.
SETTING Character personalities, skills, even dress and attitudes towards each other imply the setting. A super-hacker doesn't exist with computers to hack, students do not exist without a school, sword fighting warriors likely do not exist if there are guns and bullets. Attitudes and actions of people toward others of different gender, race, physical maturity imply a culture in which this is not severely punished.
Work from your characters to a plausible setting that could have produced all of them. If you have to tweak your characters, to arrive at both characters that please you and a plausible "normal world and culture" they come from that pleases you; do it: That is rewrite. Get used to it! Note that where they come from does not have to be where the story takes place, but where they come from can influence whether it is plausible they wound up where the story takes place, and will definitely influence their interactions throughout the story.
Pick a conflict. Any conflict. You want to write a scene with one or more characters in it, something to help you understand the characters you have until now only engineered. Think of this as a field test, we are going to write some prose, an actual fictional scene, to see how this character works, and how their personality plays into their actions.
I usually pick two characters I think are most likely central to the story, and try to write their first meeting as a conflict. How do they get together? What excuse can I come up with for them to get together? Why does A need B's cooperation; or did A save B, or inadvertently harm B, or what exactly happened? You've got some back stories, which two, when you rub them together, create the most sparks?
I have written such meetings that never get into the final story; but when I write them I write as if they will. This field test helps me discover the characters and flaws in my own thinking about them. I write several such field tests; often to see how my characters act in their normal life (which is where the reader expects to meet them). Your character design is a lot of "telling", you need to transition to "showing" how those traits influence your character's behavior and interactions with others. The process of doing that, these field tests, will inevitably (in my case) change the design, smooth out the rough edges and make the character more "real".
Eventually my goal is to have characters feel to me like they think for themselves, more like actual people I know, so when I write them what they do in response to various kinds of provocations is not really up to me.
For example, I find my character, Diane, will not ignore some lout grabbing her ass, nor will she laugh it off or walk away; and I can't make her do that to serve some plot point, it feels unnatural to me, inconsistent with her character. She's going to hurt somebody! Now I might have designed that into her, or discovered this trait through writing, but it begs for a field test: A scene where hurting somebody for playing grab ass is not going to be easy; a boardroom meeting, in a courtroom, an important client, the principle of the school threatening to expel her child. What does Diane do then? Follow through? escalate? De-escalate? I have to write a scene to figure that out, one where I consider her dialogue and conflicting emotions carefully.
I believe the right full-story central conflict for the characters is something you will discover as you develop them in prose and develop, as an author, coherency in the parts you can't really cover in design.
As Stephen King (a discovery writer) says, all stories have to turn out somehow. Once you have a clue as to the central conflict, your field tests are done. Discard them or revise them as needed to make them consistent with your characters, and put them aside until you need them. Now is the time to write your opening scenes to introduce these characters to the readers, each in their normal world, dealing with normal everyday conflicts or problems, together or separately.
You invent these everyday conflicts and they won't matter too much; they are a vehicle to carry reader interest, following a character doing something that reveals elements of their character so the reader will care about them (in a good way or bad way). Later, you may revise them, perhaps even to different events: You will be discovering many more things in writing, about the plot and characters, and when you do, you can use these opening scenes to foreshadow future events, or act as metaphors or plant hints about future conflicts. Don't worry about that now; that is something for much later.
For now, just focus, one link at a time, on a kind of chain reaction, like dominoes falling. A conflict for a character is resolved by their actions and taking steps, but the consequences create another conflict, to be resolved but create a bigger conflict. Force the characters to interact, and conflict. Write what they would do then, what is in character for them? Don't force them to adhere to your preconceived notions of what should happen. But also, don't let them walk away for long.
To this end, a writing device called The Crucible is very handy: The nature of the conflict means the characters cannot really just walk away from each other or the conflict; the stakes are so high they have to solve it.
Without a crucible, you must rely on strong character traits or extreme circumstance that would make a character endure suffering rather than violate their own belief system. Joe will risk his life and die before he abandons the search for his sister. Mary will literally kill her rich child-molesting ex before she lets the court give him her daughter.
Discovery writing will discover your plot. It is still important to understand plots and ACT structures to know how stories progress: This will help you see what you are discovering, and then refine it. Things cannot escalate forever, sooner or later your characters will reach breaking points, ultimately they will fail or succeed.
The End: The biggest problem with discovery writing tends to be the endings. To fix that for myself, I begin with some rough ending in mind, not prose but notes on how it must end. Whenever my characters start taking actions that will prevent that ending, I have to come up with a better ending, or scrap what I wrote and have them make different decisions consistent with my notion of the ending. Thus far, I typically scrap two or three endings for better endings. When I go back through the book for the second draft, I know the ending and the characters and rewrite for consistency, foreshadowing, a consistent "voice" for the characters. My third draft deals with under-imagined scenes (blocks of dialogue, blocks of exposition). My fourth is for polish; better words, better descriptions, more concision.
The biggest complaint about Discovery Writing I hear is that it is inefficient; I probably write 200 pages for every 100 finished pages. But as an author I am not devoted to efficiency, but quality, and IMO discovery writers produce better characters and better stories. To me, meeting reader expectations for a plot (including being consistent, avoiding deus ex machina, avoiding contrivance) is important, but the exact details are not. Readers come away from Harry Potter loving the characters, and hardly remember the details of the plot. Harry is a hero and wins!
To me, a serviceable plot is important but characters are what make a story loved; we love people more than we love clever situations. To me, discovery writing makes the best characters, that feel the most real, because the author built them that way from the start; letting their actions at each point be a highly plausible action for that character as far as we know that character at this point in the story. The characters end up unforced by the author, and the plot and outcome seems inevitable. I don't force my characters, exactly, I give them traits that naturally put them in conflict with each other, and I use crucibles and the 'butterfly effect' to have small conflicts escalate into larger ones: e.g. She woke up with the power out and late for work; so that could have anywhere from zero consequence to the worst consequences; pick something plausible in the middle that becomes the next issue for her to deal with and can be an escalation: a discovery, a meeting with somebody new, a new task or setback that will lead her into new territory.