I just searched on Experient (as a standalone volunteer, not with a team) and there's a room at the Renaissance Grand (essentially across the street) and the Drury Inn Convention Center.
Seems to be plenty of availability close. You need to book by Tuesday though via Experient (or so the site says). I'm at the Hilton at the Ballpark (and there seems to be no availability there)
As for the parking situation, page 12 of the
volunteer orientation has all the details. Not free, but cheap (perhaps I'm judging cheap by my NYC standards, but still.....)