There are two tunnels on Hwy 14 (east entrance) at the Bill Cody dam area - I drove my 10.5 ft. RV through with no problems - you might want to check the tunnel height before going that way - saw many 5th wheels at Yellowstone Valley Inn and thy pass...
Last year we stayed just west of Cody WY (Yellowstone Valley Inn) and drove our tow into Yellowstone (stayed 10 days). The east entrance is curvy and steep for about 6-8 miles past the check-in station. After that, it is beautiful and mostly downhill...