I've been taking the train out of the Providence almost everyday for the past 6 months. I usually grab one of the trains before 7 but you shouldn't have any issues finding a spot at 7am. The times I did take a train around 7 there was plenty of parking. I actually took a 7:40am train today and the garage was only half full.

In the one instance I experienced a full lot, which was around 10am, I parked at a garage across the street on Finance Way which is much closer than the mall and actually cheaper than the train garage.

The train parking garage is a $18 flat rate. I think it's cheaper if you're only there for an hour but I'm assuming you'll be there all day.

Hope this helps!