Use form V62 to apply for a V5C registration document:
https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/471730/V62_030815.pdf
Where you're asked to fill in the registration number, write 'Not known. No registration plates on vehicle when purchased'. Enter the tax class as 'Motorcycle'. Importantly, make sure that you enter the VIN (or chassis/frame number) shown on the moped. Fill everything else in, send the form off and wait to see what happens.
If the DVLA can trace your vehicle from its VIN (or other number), you should get a V5C showing the registration number. If they can't, expect lots of questions!