I just got done running the Yampa, definitely my favorite desert river. It is so beautiful. Anyway, I did a little research for you at things are looking bleak my friend (or possibly bright?)
If you were to talk to Kelly (head at the river office) she will tell you that 1000 cfs is her recommended minimum. However, I know people that work on the river doing work with the endangered fish and they'll tell you down to 800 is fine (mind you all these numbers are for rafts). And, most recently (last week) I talked to a ranger who's name I'll not post who told me he's done it down to 650!
All that being said, I think you're looking at 1000-600 cfs on launch day (and falling of course), which though I just said the ranger said is doable....I personally wouldn't recommend if your group is very big at all. I think the ranger is able to do it so low because their boat is only packed for one person vs. a raft full of people and beer
But...i wouldn't cancel your trip. That's what everyone else is going to do. If the river really does run that low, most if not all permits will be canceled. Commercial trips are definitely not going. Meaning you can have the entire canyon to yourself. I would plan an epic ducky trip! You can rent them for relatively cheap and maybe given the amount of time you have round some up from friends.
I've done the yampa twice at low flows ~2000 (last July 31st crazy) and recently around ~3500. I would do every rapid at these flows in a ducky (especially with the warm water you'll have in June ~70)....so given that you probably won't even have half those flows I'm sure you'd be just fine. Ducky trip....that's what I'd do, it would be so much fun.