Ok, so, let me make one thing clear: parrots, regardless of the species, are NOT emotional support animals. They are difficult to care for and will need a lot of attention (3 or more hours a day). My birds do help me, but keeping them can get REALLY stressful at times. I recommend searching up "reasons you shouldn't get a budgie" or something like that, and then see if you still think it would be a good idea.
I wouldn't get a bird without having a trustworthy vet in the area. But if you're just unaware of nearby vets, I think you could just do some research and find one.
Many parronts live in areas of the world where there are no avian vets, but I'm not sure how it works out, so I can't speak for them.
About the car ride. Eight hours seems like a really long time. The thing is, every bird is different, so some might get stressed out, and others won't.
I think it should be ok, but you can't do it regularly.
Also, what's your plan once you get home? Are you going to have the budgies stay in their carrier (except for daily playtime, of course)? Are you going to bring the cage with you? Are you going to get a small cage to begin with and have the birds use it as a sleep cage? (I don't think that last option could work out in college, but I don't know.)
It might be possible to do a lot of research and think of a plan that could work out, while making sure the birds are happy and healthy, but that's gonna be hard. You might wanna hold off on getting a budgie until you graduate.
(it's your decision, though. If you strongly believe you can do it, I won't say no to that. I'm just offering my opinion.)