Having been in two long distance relationships before, both ended badly. The distance really kills it. Plus you barely know her to begin with.

There’s this girl who I have an extremely tenuous line of contact with who I would like to become better friends with

So do it
Simple as

Chat them up and see where they are at in life. Test the waters to see if they are interested in having the kind of relationship you want to have. If they aren't, which may very well be the case, then move on. Don't become hyper fixated on it or try to change their mind, just move on. Be intentional about the type of relationship you want to have, don't pretend to be a friend then push for more. You'll save yourself a lot of time and anguish if you're honest with yourself and honest with others.

Are you planning to move there anyway? Is she planning to move where you are anyway? In my experience crushes always intensify after it's too late to pursue them. If you don't even remember if you were in the same year in school, it seems to me that you don't know each other well enough to justify some romantic journey across the world. By all means, contact her if you like, but be up front about your intentions very soon and be prepared that she's not interested. Don't torture yourself by chatting on and on without getting to the point.