No, you can not wear a name tag.
From the Auxiliary Manual, COMDINST M16790.1 (series), Sec A.2. Vessel Examiners (VE):
For VEs, the VE Polo shirt is authorized for wear as an optional uniform shirt worn as part of the Undress Blue Summer (Bravo), Working Blue, ODU, and Hot Weather uniform. The shirt is not authorized for wear on patrol and no insignia, name tags or breast devices shall be worn with the shirt.