Yes, Granger is a Mudblood. She seems proud of it, from what I've observed. She was taken in by Sirius's band of misfits almost immediately. To her credit, she didn't seem particularly willing. Very little is known of her past. When asked, she doesn't even come up with a lie, she just refuses to respond. There is a rumor going around that she's a refugee of war, though I don't know enough to prove or disprove it.