I have updated the wildflower guide for Mt. Rainier NP with a new and expanded description and map of the wildflower areas. The wildflower season isn't for another 3-4 months, normally mid-late July, earlier with a low snowpack or early snowmelt or later with the opposite, a high snowpack or later and longer snowmelt.
Last year the snowpack was far higher than normal and the snowmelt the longest on record which resulted in a very late season, early to mid August in low to mid level elevations and late August in the upper elevations. This year similar as the snowpack is above normal so far, but the peak doesn't normally occur until early-mid May and the end of the snowmelt won't be known until late June to early July.
The wild flower season normally lasts 2-3 weeks so your time is very narrow, but it's dependent on the snow and the other factor which influences the wildflower season after the snowmelt, which is the weather in July. A cool, rainy July will delay the start of the season as the plants won't begin to bloom until the weather clears and warms.
So when it comes to your visit to be there for the wildflower season, it's in the old adage, "Timing is everything" and all I can say is good luck and watch the news and reports.