So I'm wondering if this supporting character, who I use for exposition purposes, can be made more vital in some way. When does a supporting character become necessary for the plot? What criteria do we use to judge that?
When does a supporting character become necessary for the plot?
Think of a supporting character (SC) as a partner to the main character (MC), somebody they can rely on when they need help in a jam they truly can't get out of themselves. The MC gets captured, or injured, or drugged, or arrested, or even confused: And the SC has the solution.
The SC is necessary to the plot when you engineer some situation in which the MC would fail without the help of the SC. This "help" can be physical, emotional, analytical, or all of those. It can be the SC warning the MC of a danger the MC did not see coming. It will help if you try to think of "supporting character" as actually providing some kind of necessary tangible support to the MC.
Of course the alternative is that the SC just fills some gaps in the story, provides the author an excuse to revisit memories of shared experiences, etc. They can be a foil to help you convert boring one-sided exposition into conflicted conversation (Jack laughed, "That's a total lie and you know it!"), ("conflict" means emotion rousing, so it can be humor, laughter, sadness, nostalgia, anger, frustration, etc).
This doesn't make the plot dependent on the SC, but it makes them a necessary character for the story to unfold smoothly, and feed the reader information without having to make them stop and eat their spinach with background exposition about why the MC became who she is.
A good SC can perform both functions, even if they have never met the MC. As a friend, they can invoke shared memories. As a stranger, they can ask questions. Just make sure these conversations feel like conversations and have some emotions in them and disagreements or misunderstandings or frustrations or whatever; the characters should be feeling things. Otherwise it comes off as boring exposition anyway.
However the SC saves the day, that is something you, as a writer, must build into the story from the beginning. In order to avoid this SC help at a critical moment seeming like a deus ex machina, you must give the SC and MC some reason for the SC to be around, and the MC to let them be around. They can be friends, of course, but show us some reason they are friends, the SC makes the MC laugh, or the SC is not hero material on their own, but really does often have better ideas on how to proceed than the MC does, and the MC doesn't mind that, or even knows they are not genius of this group.
A SC should walk a middle road. You need to make sure the main character stays the hero of the story and makes the final decisions and takes the final risk, but you can make the supporting character the hero of the MC, if that makes sense. The supporting character finds a way to get him out of jail or frees him when he is trapped, or distracts the enemy, or defeats the alarm system, so the hero can infiltrate. The SC is the one that realizes they were betrayed, or that they are falling for a trap: The MC is the one that does something about it.
But I say a middle road, because although you don't want the SC to be the hero of the story, you also don't want the SC to be just another burden the MC has to constantly rescue. You don't want your story to be about something the MC could have done easily alone in an afternoon, but to get it done she had to struggle for a week with a grown man baby strapped to her back.
It is generally also considered cliché to write a beloved SC that is just going to be killed or kidnapped in order to motivate the MC. This is often done as a love interest (wife, child, girlfriend), lifelong partner/mentor (cop on his last day before retirement, oh no!), or even just a helpless friend.
To be necessary to the plot, they must be necessary to the MC, which means they must have a talent or ability the MC does not have, without which (and you make it so) the MC would fail in their quest or mission.
A character need not have any impact on the plot, in terms of making things happen, to be an active player in the narrative as a whole. Take S.M. Stirling's Odard Liu; he spends three books as a minor character who serves as a reminder of where he comes from and something of a punching bag of fate, being at the mercy of the unscrupulous actions of his mother who is hundreds or more miles away, then he dies. His plot impact is zero, he doesn't make anything happen, though he could be said to help in several cases he's not necessary. So what does he do? He serves to illustrate two points:
no man is all or one thing.
people can and do change.
Both of which are important themes in the work and his progress as a character parallels, but is also rather different to, the journey of the protagonist.
Odard comes to mind with this question particularly because he does not get along with the main character of the tale for most of his personal story as is in fact generally not a particularly nice, or good, human being.
Characters are only necessary to the plot when they carry out a task, or tasks, that cannot be completed by another character and those tasks and the events around them drive the plot forward and they are essential to the continuing progress of the narrative. Particular individual characters are very rarely necessary to the plot; most times there are other ways to resolve events either by using a different character, or changing the details of how events play out, to get the same result. At best you can say that a character was necessary in a given role.
Personally, I would scrap any character who does not impact the plot at all.
If you remove that character completely, would the plot stay the same? If the answer is 'yes' I would either scrap that character or find a way to make them more vital to the story.
If a character exists solely for comedic relief, consider combining that character with one that is more important and together they'll create a more concrete/interesting character.
To make a supporting character more vital, give her her own arc. How does she grow and change over the course of the story? What does she really think about what the MC is doing? What backstories does she bring to the table?
You don't have to use these things explicitly, but have them in mind. Have them inform the small things, like how she responds to the MC. Details like the expression on her face, the couple words she says, her body language, and her response time will reflect her inner state. With enough of those tiny bits, your reader will have a better sense of her. She will start to feel like a real person and not a device.