Many trails are wheelchair-accessible. There's a couple guides on the web, specializing in the low vs high country (4 season vs summer season) trails. Here's one:
http://accessibletrails.com/Olympict...ympicindex.htm
Best season: May and June for wildflowers. The season starts in April with at low elevations and moves up the mountains, of course. Rhododendrons are at low-to-mid-elevations, 1000 to 4000', so run from April through early June. The alpine wildflower (Hurricane Ridge, Deer Park, etc at ~6000' MSL) season begins in late June, depending on how much snow we got the previous winter, and how warm the spring is. They appear in greatest abundance just as the snow melts off. Many of these species are quite tiny, and it's nice to have a hand lens (or a macro lens for photos). The alpine lilies and paintbrush peak in July.
The Olympics can get rain anytime through typically mid-June. Factor this into your plans - if your tolerance for rain is high and crowds is low, come before mid-June, and vice versa.