As much as some would not like to see and more so hear snowmobiles in Mt. Rainier NP, there are places where they are allowed to share the road with other winter travellers. As noted on the NPS Web page (near bottom), you can use a snowmobile as follows.
"In the southwest corner of the park, snowmobiles are permitted for 6.5 miles along the Westside Road from its junction with the main park road as far as Round Pass. Beyond Round Pass, the Westside Road is closed to snowmobile use. Snowmobiles are also permitted on all the road loops of Cougar Rock Campground. The campground is closed to overnight use during winter and the roadway is left unplowed. Contact a park ranger at the Longmire Museum for maps and additional snowmobile information."
A word of note.--The Westside Road is blocked at the Fish Creek trailhead just past Dry Creek. It is not advisable to travel beyond the barriers across Fish Creek at the trailhead and on to the second crossing of Fish Creek if the conditions aren't reasonable for snowmobiles. The trail/road between the crossing is often damaged by floods and not maintained beyond a basic 4WD road at best.
"On the north side of the park, no ranger station is open in the winter. The US Forest Service District Office in Enumclaw provides information and maps for White River, Carbon River, and Mowich Lake areas. For more information, call the USFS District Office in Enumclaw at (360)825-6585. Highway 410 is closed near its junction with Crystal Mountain Ski Area road, at the North Park Boundary.
Snowmobiles are permitted on the 12-mile section of unplowed road from the North Park Boundary on Highway 410 to the White River Campground. Snowmobiles may not continue on Hwy 410 south of the White River Road turnoff. They are also prohibited from proceeding beyond the closure at the White River Campground road junction towards Sunrise. Snowmobiles must stay on the road corridor; they are not allowed to proceed beyond the campground towards Glacier Basin. Be aware of avalanche danger and the Mount Rainier weather forecast.
In the southeast corner of the park snowmobiling is allowed on the Stevens Canyon Road from the Stevens Canyon Entrance to the road tunnel at Box Canyon.
Wilderness permits, required for all backcountry camping, and climbing registration cards are available at the north boundary arch on Highway 410 or by self registration at the Ohanapecosh Ranger Station."
Personally I could do without them but there are places there are useful and helpful. I only remind snowmobilers, as they already know, to follow the rules, stay on the allowed roads and share the road appropriately especially in areas of limited view or in adverse weather. Those of use on foot, snowshoes or skies aren't as quick or fast.