I am unable to submit online quizzes on time due to technical difficulties, and it is negatively affecting my mental health. Can I get extra time?
About your actual question, yes, there is a way to get extended time based on having anxiety during exams or related things (although I cannot make any guesses as to whether you would qualify given your specific situation). Here is what I think is the relevant office at your university to consult about that.
About the broader context to your question, this situation is very sad and unfortunate. I think your instructors are probably unaware of how badly their tech cluelessness is affecting their students. If they are aware, then treating you and your fellow students this way intentionally seems unconscionable to me.
Given this, I think you should consider going not only to your professor but to someone in your program with the story of how this situation is creating an impossible learning environment for you. I’m sure there are sympathetic, well-intentioned people at your program (e.g., an adviser, undergraduate program chair or vice chair, etc) who have authority and would be able to advocate for you, and would genuinely want to know about and address these types of problems, not just for your sake but for all the students’ sake. I can tell you that just as you are struggling with studying in an online environment, many professors are equally struggling with teaching in such a setting as well, and do not always have everything figured out about addressing the many practical and logistical challenges that come up. Hearing from students about what is working and what isn’t provides vital feedback that would enable them to correct course where appropriate.
Good luck!