Can a professor grade students based on what she "remembers" with no proper record?
Check if your school has a grade appeal process.

Many schools in the US have one. It will bring in some other administrators (e.g., chair of the department or dean) to review your argument and assess the professor's rationale for giving you the grade.

You might have a good chance with this since she has no written justification for your grade or other student's grades and relies only on her memory. Additionally, if you can remember what you spoke about during the class on specific days (and the students who spoke immediately before/after you), you might want to go ahead and write that down so that you can bring it up with your appeal.