Do you need a cdl to operate a bucket truck?

Depends on what the gross vehicle weight rating of the truck itself is. If it's under 26,000 lbs, then no.

Single axle trucks can be a bit ambiguous as far as this is concerned. One example would be the Ford F750, which is built to be a 33,000 GVWR vehicle, but is marketed as a 26,000 GVWR vehicle. So long as the weight of that vehicle remains under 26,000 lbs. GVWR, it doesn't require a CDL.

Tandem trucks and larger, however, are not so ambiguous - you need a CDL to drive them, no matter what.

Unless it is exempt.