Yes, it can be done with local anesthesia or without anesthesia, but it is not recommended. Doing the puncture without anesthesia implies pain, quite strong in the case of a high number of follicles. In addition, it is more uncomfortable for the specialist and, therefore, the intervention can be prolonged and lead to greater complications.
With regard to the use of local anaesthesia, such as epidural anaesthesia, the recovery time is longer and is therefore not usually used.