Is it OK to ask your supervisor for a recommendation for attending an academic event if the PhD just started?
Rather than just ask for the support letter, it may be a good step to simply ask if your supervisor thinks that the event is appropriate for you to participate in, given your knowledge of the topic and ability to prepare for the event (if applicable, I'm not sure if the event is something you would have to prepare for, like a conference). Your supervisor may be happy that you are demonstrating such enthusiasm and effort early in your program of study!

Since the deadline is short, I would offer to draft the letter for him. This will help with the timing, plus it will help since you have not worked together very long. This will ease the burden of the task, which he will also appreciate. Good luck!