My boyfriend's US residency case is pending. He has lived in the US for a long time and been a California resident (He has his CA driver's license) for a long time as well. Is it OK to travel to the US Virgin Islands?

It depends on the stage of the case and his immigration status prior to filing. Does he have an advance parole travel authorization issued out to him?

No he doesn't.

What stage is his case in and what is the basis for his PR application?