1. This is what I know: In British English, you can only say the second one - you have to include the 'to'. ( You can say 'I'm going to write you a letter' but not 'I'm going to write you') In American English, you can say both 'I'm going to write to you' and 'I'm going to write you'. 2. This is what I don't know: I don't know whether the version with 'to' is considered more 'proper' among AmE speakers than the version without. It would be interesting to hear an AmE speaker's perspective on this. 3. This is something you didn't ask about, but I'll say it anyway. Don't write 'gonna' unless you are imitating the way that somebody is speaking ( or singing). Even in informal contexts, it's normal to use the standard written form. This is 'I'm going to'.

"I'm going to write you a letter" write + (to who) + the article + letter/note/memo etc "I'm going to write to you" write + (to who) both the same it is never "I'm going write you" something that is seen thousands of times a day on italki

I think both are incomplete, but when talking are understood. In writing, "gonna" just isn't done. If you want to shorten the sentence, I would say "I will write to you" and with contractions "I'll write you".