Longmire [1] has a sit-down restaurant inside the National Park Inn (360/569-2411, open year-round). The dining room at Paradise Inn (360/569-2413, late May–Sept.) serves three meals a day and a big Sunday brunch during its season of operation.
For something less formal, the Paradise Visitor Center (open daily May–early Oct., weekends and holidays year-round) operates a snack bar with typical fast food.
The cafeteria at Sunrise Lodge operates only between late June and early September. Limited groceries are available year-round at Longmire General Store and during the summer at Sunrise [2].
Call ahead, as hours vary from month to month at all of these venues.
Links:
[1] http://www.moon.com/destinations/washington/south-cascades/mount-rainier-national-park/sights/longmire-area
[2] http://www.moon.com/destinations/washington/south-cascades/mount-rainier-national-park/sights/sunrise