I wouldn't expect a lot of flowers until the snow has had a chance to melt. Check out the second picture on this current post and you will see that a melt is a ways off yet.
As for Sunrise, the park service always try to make a point of opening it up just before the July 4 holiday.
I was down near Rainier this weekend and there were plenty of flowers below the 3500 foot level, so you just have to stay in the woods. The Ohanapecosh and Longmire areas are your best bets within the park.
Here is the link for the Mt. Rainier National Park Web cams:
www.nps.gov/mora/photosmultimedia/webcams.htm
We took a drive up to Longmire today, a few flowers along the road at lower elevations, and the fragrance of Skunk Cabbage in the air. Wild flowers at Paradise and Sunrise are more late July and well in to August, still snow up in those higher elevations right now.
Rich T.