It isn’t impossible, but it is pretty unlikely. For starters, it would depend on availability. You’d have to wait for juniors and seniors to reserve their rooms and then see if there are any left over. (Note that juniors aren’t even guaranteed singles — although I’ve never heard of any not getting one.) Regardless, what makes this difficult is that it’s such a gamble: while you are waiting to hear about all this, you can’t exactly have the back-up plan of already having a room and roommate secured, because then what would your roommate do? And also, everyone would do it if it were easy. Having said all that, I know a few people who have done it.
Also note that with a note from a doctor you can get a medical single. House Chairs also get singles — even if they are sophomores — but that shouldn’t be your reason for applying to that position.
You can ignore everything I just said if you want, because what you should really just do in this situation is email Sam or Sage (our housing goddesses) and talk it through with them. Especially if you are a current student, which I have a hunch you are. That way they can look at the nuances of your situation (ah, the perks of a school of 700).