While this is an unusual way to conduct quizzes in an academic environment, it is letting you practice some new skills in time management and risk management, so I do not share the strong opposition of others to the practice. I also think that academics should have wide latitude to set the rules of their assessments, and while it's not how I would do things, I think this probably falls within the scope of reasonable discretion. (Whilst there may be some "tech cluelessness" at play here, as Dan notes in his answer, I disagree with his strong negative view of the practice.) I also note that the constraints seem to be the same for all students, so presumably other students are also having to manage the time required to download the quiz and upload their answers. This effectively just means that the total time available for the substance of the quiz is reduced for all students.
In any case, in view of the difficulties you are experiencing, I would encourage you to form a realistic view of the constraints you are under, and formulate sensible risk-management processes as a response to these constraints. Here you have a situation where there is a variable time cost for obtaining the quiz and for submission, neither of which you can perfectly predict in advance. There is also an extremely heavy penalty for late submission. That set of constraints means that a good risk-management process would be to give yourself a substantial "buffer" (e.g., five minutes) by submitting your quiz early, even if the quality of your submission is poorer than it would have been if you worked a few more minutes on it. Giving yourself a time buffer will give you some additional time for uploading if things go wrong, which reduces the probability of failure to submit within the allocated time. From a risk-management perspective, the "buffer" method reduces the quality of your submitted work slightly, in order to substantially reduce the risk of getting zero on the assessment. If the buffer is set to a reasonable level, this should give you a higher aggregate mark.
While it is unfortunate that you have already lost some marks on this, it tells you something useful: your time management and risk management skills are presently not good. Risk management and time management skills are important to develop during your education, so I think you have an opportunity here to identify a deficiency in your skills and work to plug it. For your next quiz, set yourself a buffer of five minutes, and aim to successfully submit your work no later than 1:45pm. If you can accomplish this, then set a goal to submit all future quizzes successfully within the time constraint, using whatever buffer you judge to be prudent. If you can look back at the end of semester and see that you managed to improve your time management and risk management on this process substantially then you will have formed a useful skill for your later career, which is something that will be useful, and of which you can be very proud.
Now, to deal with your main question, you can certainly seek additional time if you like, and there are usually formal university procedures for doing so. You will need a justification for your request, and the mere fact that you are under a difficult time constraint that also applies to other students would not suffice. Also, negative consequences to your "mental health" (by which you mean, essentially, that you are experiencing stress) caused by the existence of a tight deadline and your poor time management of that deadline, is highly unlikely to give you a valid disability claim. Based on your question, I see no valid grounds that would justify offering you a time increase. In any case, I would encourage you to try to adopt better practices here rather than seeking extra time --- give a man a fish and you feed him for a day; teach a man to fish and you feed him for a lifetime.
