Can I request references from people in the program I'm applying to?
I think that would be a great idea: your references are supposed to be advocating for your ability to perform the work of the program, right? Well, who would know better than people running the program? 

If one of the program faculty gets asked about you over coffee in the morning, and they say "Oh, AS2045? Yeah, he was in my undergrad classroom, and I think he'd be a great fit for our program, bring in a ton of good grants," then odds are, you're going to do well there.