"She does not run a lot."
あまり (amari) is used when you're saying you dont do something often, or not that much (for example not reading often, or not sleeping that much). たくさん (takusan) is used when you're saying there is a lot of something (for example saying there are a lot of books, or there are a lot of people.)
I think that because you are talking about "her" without using her name, you are assuming the person you speak to knows who "she" is, therefore, "she" (who ever she may be) is within context, and so not a new subject introduced into the conversation. Because of that you use は. If you referred to someone/something now previously mentioned during the conversation, then you would you が.