The (small) pup is extremely friendly. He’d love to come along for the ride and daughter would like to have him there. Would there be a problem with bringing him? Would it seem weird?
I do not know a thing about Vassar or the protocol for move in but just wanted to put in my 2 cents. I think it’s a bad idea. While your pup may be friendly this is going to be a chaotic event. There will be tons of people, lots of activity, boxes and stuff everywhere. This can cause anxiety for even the most friendly and outgoing pet. Plus you may not be able to bring him in any building and would have to leave him in the car. Temperatures can rise to dangerous levels and he could suffer from heat stroke. Plus with all of the activity he could get loose and lost. I say leave him home.
I’m not sure the school would like that - while your pup is small and friendly, if lots of parents started bringing pets to drop-off it could be a huge problem. Move-in is very chaotic.
D does love it when parents bring dogs to visit during the year - but that’s usually on a random weekend when there isn’t as much going on. I brought her dog once, but that was a quick pickup and go, we were only there 1/2 hour or so.
Vassar has faculty that live on campus that that have dogs. One always see dogs around campus. That being said move in day is very hectic and there are many events that you will want to attend. Who,will watch the dog when the family is at the welcome ceremony? By the way , if your child is from an animal loving family like ours, Vassar is one of the few schools that allow pets. My son has had a canary during his time there. You are allowed to have a pet that you can keep in a small cage.
Aside from the move-in day issues, your dog would have a blast there. There’s lots of room for running around on the farm, by the lake etc.
I would only bring him if you have a person who can be dedicated to watching him, maybe taking him to a grassy area to play ball while the rest of you move your daughter in. It would be terribly inconsiderate to bring him into the building while people are trying to move and the roommate is trying to unpack. Some people do have dog allergies as well. Bringing him also makes going to a local restaurant difficult. However, if someone can watch the puppy and keep him out of the moving in chaos, he might be a real stress buster and tons of people will probably want to pet him out on the quad.
I have a very portable 7 lb chihuahua that the D’s and I love. I thought about bringing him with us when we dropped off each D at summer camp (at a college). Once I thought it through I realized that if I brought him, he becomes my primary responsibility because he can’t ever be left to his own devices. Because he’s a dog 
I couldn’t leave him in the car, if he had to go to the bathroom, I had to take him out. I would not have a hand free to help move stuff in and organize.
In short, for that day I needed to be able to be available for whatever my daughters needed, and if I had the dog, I couldn’t do that as well as I could without the dog. And I really wanted to be able to focus on dropping them off and picking them up and all that went along with it.
So we boarded him at his favorite boarding place. When we were picking up D17 there was a mom who had brought their dog, and she was sitting with him in the shade, waiting for the rest of the family to pack up. It definitely felt like the right decision to me not to bring him because I felt like I would have been like that woman-missing out on what her kid was doing and just sitting under a tree waiting.
Maybe you can check with the housing office?