Yes
Yellowstone is "open" year around. It's just not all the roads are "open". If the roads are closed the buildings may also be closed, and so are the campgrounds.
The only road open (to vehicles) is the north entrance at Mammoth.
When the rest of the roads open depends on the year, a "low snow" year it could be April. A heavy snow year, well, I've had a snow-ball fight (from fresh fallen snow) on July 4th. I've seen years when the passes aren't opened until mid June.
On average it snows (a little) every June.
It really depends on the year, most El Nino weather patterns lead to a low snow pack (in 1977 I drove through the entire park in early April). So this -might- be a low snow year. But maybe not....
I used to live in the area and have been in the "park" every month of the year.
