Upvote:6
No. Emergency visa appointments are for "urgent medical or humanitarian need to travel" (source: US Embassy in the UK). Your father has no such need: there is no medical situation and if your internship in the US creates some kind of humanitarian problem, it's a problem entirely of your making, so the solution to it is that you don't travel to the US. An emergency appointment would be denied.
Furthermore, your father should only re-apply if he has fixed the problems that caused the first application to be rejected. Don't think that applying again gives you another chance to get lucky: they'll see the first rejection and the first thing they'll check is whether the same reasons for rejection still apply.