Can chin augmentation under local anesthesia be done while pregnant?

It is generally recommended to avoid any elective surgery, cosmetic or not, while being pregnant.  While cosmetic surgery is relatively safe, there are risks, and it is better to avoid any of these risks affecting the fetus in a pregnant woman.  I would recommend waiting to have your chin augmentation until after your pregnancy and when you have finished any possible breast feeding.  

Elective surgery during pregnancy is generally avoided. In the unlikely event that there were complications, for example an infection, then systemic antibiotics would be required and and in some cases these antibiotics have been known to affect the developing baby. I would recommend to wait.  

Purely elective aesthetic surgery of almost any type is not going to be done on a pregnant patient. While the risks to the developing baby may be small, no one can say it is absolutely zero risk.