Yes it is possible to do a tummy tuck under local but it depends on your preoperative findings and your personality. Discuss this with your doctor.
If the patient is comfortable then the surgeon is comfortable! A tummy tuck involves excision of a large amount of skin and fat, tightening of the abdominal muscles, liposuction and about 45-60 minutes of stitching things very carefully and precisely. At the half way mark, your bed is changed into a V shape so that the skin can be closed safely and tightly. You would likely reach a toxic dose of local anaesthetic to numb everything properly. A spinal or epidural anaesthetic would also be quite tricky given the position changes during surgery. The risks of general anaesthetic in an otherwise healthy patient are extremely low. Maximise your safety by having your surgery performed by a board certified plastic surgeon who uses an equally qualified anaesthetist.