It is possible to find the Registration from the VIN (frame number). I have a Trade HPI account which does it.
I type in the VIN and the report comes back with any information held on the vehicle. If there's no information on that VIN then the result will be "No Records Found".
A couple of points to note:
1. The check runs on the exact number entered - any mixed up/un clear numbers will give a false result.
2. All records held for the vehicle will be returned; make, model, year, reg number, accidents, theft claims, theft reports and sometimes some other bits.
A word of caution. If I HPI a vehicle and it shows up as stolen I'm obliged to call the Police Station dealing with the incident - all other people with a HPI trade account have to do the same. If the bike has been reported stolen and is on the police database they will come asking. If you bought the bike in good faith it's unlikely you'd be charged but the bike could be confiscated.
DVLA, the police and normal HPI check companies won't do a VIN check.