You can get more information on snowmobiles in Mt. Rainier NP and from the NPS Website, which is 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 Information Center for maps and additional snowmobile information.
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 weather forecast.
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.
I would only suggest that you ride carefully and watch out for snowshoers, skiers and hikers. They're on the same trails and roads too and they can't react as fast as you and you can often see or hear them.