I believe the snow cats are for staff use, I could be wrong but I don't believe you can hitch a ride on one of those.
You can get on a Snow Coach though. This is the 9 passenger vans from Great Glen Trails that has been converted to a 4 wheel drive track system.
From the Mount Washington Auto Road website:
"SnowCoach tours are approximately 1¼ hour long and depart daily from December through March from Great Glen Trails base lodge at the base of Mt. Washington on a first come, first served basis. Operating hours are approximately 8:30 am to 3:30 pm, depending on weather and snow conditions.
The cost for this winter sightseeing experience is $45 for adults and $30 for children ages 5-12. Advance reservations are not available due to unpredictable weather and snow conditions."
Sounds like a great experience... good luck!!!!