Hi there! I think that it's always a good idea to get a budgie checked out by your avian vet when he or she comes home - in this case, I think it's a she as well, but it will be easier to tell in a front photo of the cere without flash.
Regarding the vent, the discoloration of the feathers might have been caused by the bird going to the bathroom and the essentially sitting close to the droppings. Since it looks like there aren't a lot of perches in the Petco photo, and they aren't very high, this could be part of the problem. Your bird will also be stressed for a while at her new home, so it's normal for droppings to be a bit more watery, but she should not be puffed up, overly sleepy, or display any other signs of an ill bird.
I would put her on a probiotic immediately (I use Harrison's Booster) and start putting seeds or pellets - whichever you choose - and fresh fruit and veggies everyday. Some egg food will also help her energy during this stressful time of coming home.
If you don't see improvement within a week, or if you notice any other symptoms of illness, I'd take her to an avian vet immediately.
Good luck and I hope that she will adjust quickly to her new home!