A section of a story I am writing is being told in third person from the perspective of a character who does not know anything about herself due to amnesia. The reader is also unaware of anything regarding this character, but she assumes an identity after a period of time based on the things she learns. How can I refer to this character in 3rd person until her identity is assumed?
Use the name others use for her.
It's pretty standard that, if a patient can't be identified, a placeholder name gets assigned. Jane Doe (in the US anyway) is a very common one (John Doe for males).
If this continues longer than a few days, the hospital staff (or the people wherever she finds herself) will come up with a nickname for her. Or your main character may make up a name for herself. Sometimes this is just a first name. Other times it may be a first and last name, for legal purposes.
If you don't want to show the reader any name, you can refer to her as "the woman" or something similar. Some stories do that. It's not my style, but it works.
Or you might use a nickname that isn't a real name. For example: "5B" (the room she's in) or "Yosemite" (where she was found) or "Redshoe" (what she was wearing when found).
Humans always assign names to people, even if they're not accurate, complementary, or real names. Give her something that has little to nothing to do with her real identity and go with it.