I give this site credit for improving my grammar.

One could also rephrase the above sentence as

I credit this site for/with improving my grammar.

Should the word be "for" or "with"?

Thank you.

Use "with."

PS - I notice that this is only your 8th post - Welcome to the forums.
Thanks, Grammar Geek! As you may have noticed, I've been posting all the questions I've saved up over the days/months/years.