Is It Possible to Get a Sliding Genioplasty Even Though I Have an Underbite?

There is no connection surgically between a sliding genioplasty and the position of your teeth. The part of the jawbone in which the sliding genioplasty is done does not affect the part of the jawbone where the teeth are.

Yes, it is possible.  However, I would need to examine you and see if a sliding genioplasty would benefit you from a cosmetic standpoint.  TYpically, patients with an "underbite" - where the lower teeth are in front of the upper teeth - would make the cosmetic appearance worse if the chin is further emphasized.