Is it ok to asking several profs at the same time for a letter of reference on short notice? I got an offer for an exchange program and I have paid the deposit, which isn't cheap. The application deadline (which I was only informed a few hours ago) is due in two weeks, on 2nd March. I have to get a letter of reference from a professor by then, but it's hard to contact them right now because public holidays (lasting for around 6 days) are starting on the day after tomorrow. If I do it after the holiday then I'll have less than 5 days until the deadline. I have emailed a prof, asking for an appointment but haven't got back any response. Is it ok to ask a few more profs in case I don't get any replies or get rejected by said prof?
Because of the workload professors are under, it is not an uncommon problem. The only way to address the problem is to contact multiple professors for a recommendation; I know other people who have contact multiple professors for this purpose.

You have only asked the Professor if they would be willing to write a letter of recommendation. This means that they have not already invested any time. So, If you get multiple recommendations, you can simply email the other professor(s), "Thank you very much for agreeing. Another professor has already offered to write me a recommendation letter, so to save yourself some time you don't need to write one for me too".