Can the bailiffs take a vehicle that's not in my name but the debt is in my name?

No

The bailiff may take control of goods only if they are goods of the debtor.^[1]

The named person on a vehicle’s V5 is not necessarily the owner, and the front of the V5 document states:

THIS DOCUMENT IS NOT PROOF OF OWNERSHIP

It shows who is responsible for registering and taxing the vehicle.

If bailiff takes your car for someone else’s debt, then you can make a third-party claim.^[2]

^{[1] Paragraph 10 of Schedule 12 of the Tribunals, Courts and Enforcement Act 2007
[2] See Making a third Party claim}