Turkey Neck Procedure for Diabetics and Under Local Anesthesia?

Depending on the extent and complexity of the procedure as well as your tolerance for the injections, a simple cervicoplasty can be accomplished under local anesthesia with a relatively low complication rate if you are a well controlled diabetic.

A neck lift is best done in a controlled setting under anesthesia.  I would not want it done on  me under local only.

The ideal neckline is a 90 degree angle on a side view while the patient gazes straight forward with a visible jawline & no excess hanging skin or fat. The neck should have a smooth surface contour without horizontal skin folds or vertically visible muscle margins. The skin should be supple not stretched tight or hard.
The treatment for deviations from this ideal depends on the cause of the deviation. The ideal candidate for liposuction is some who has good skin tone, most of the fat in the neck between the skin & underlying muscle rather than under the platysma muscle of the neck, little or no neck muscle laxity & appropriately proportioned/prominent jaw bone & chin. The result is less dependent on the amount of fat removed than on how much that fat contributes to the preoperative appearance vs. lax musculature, recessed jaw bone etc..
One good way to tell is to grab the neck skin & fat between your fingers while biting down & gritting your teeth. This tightens the neck muscles and gives the trained eye an idea of muscle laxity, fat location... However, even the most experienced eye sometimes has trouble with this assessment manuever for example when a lot of neck fat is present or when the muscles are extremely lax.
The treatment for very prominent or lax muscles & very lax skin is a facelift procedure. The name is a bit of a misnomer as the neck is often affected more than the face. This allows removal of excess skin & tightening of the muscles when needed. The surgical approach varies slightly depending on sex, age, other physical factors, etc.. The treatment for recessed jaw bones is dependent on the severity of recession & dental alignment. If the teeth are aligned properly & recession is not great an implant can be placed via an intraoral approach to make the jaw bone &/or chin more prominent. In the process the neck profile assumes more of a 90 degree angle in appropriate candidates. In a minority of patients the neck can be treated by liposuction & surgical tightening of the neck muscle alone. This results in a quicker recovery time with lesser chance of complications than the classical facelift procedure.
Getting back to your question whether or not local anesthesia alone will suffice I can say that depends on which of the above procedures is required and what your physical condition is.