Is it okay to write a letter for someone while trying to give them as much space as possible?

I was trying to talk to a teacher during recess today, but she responded with a slightly harsh “no”. Then, during lunch, I asked her if she was going to eat, and then she came to me and told me to stop waiting for her and saying things to her other than “hello” and “goodbye”, as she was very busy and pressured—to be fair, however, I was just curious, nothing else. Now, I am perhaps thinking of writing a letter for her but giving it to her around next week, just so she can understand what I want(ed) to say. Is that alright for me to do that? I ask this because I don’t mean to come off as a nuisance to her or make her feel uncomfortable.