Sorry to be the bearer of bad tidings, but I phoned a decent hotel I know of (
Hotel Sautter) to try and find out about availability for sometime next June for my brother and sis-in-law, and they're fully booked for the whole of June and first half of July, and I think you'll find all other hotels will be the same.
Your best bet is probably to look further out of Stuttgart itself for somewhere that's still on the u-/s-bahn network and you could still get quickly into the city using public transport. Unfortunately, I can't give you any recommendations though -- you'd just have to try pot luck on the internet.