Make sure your vet is an avian vet, most veterinarians have limited bird knowledge, so you've got to make sure it's a vet that knows birds.
In my experience, taking a chick to the vet was a waste of time and money. The visit was $95 and in the end the avian vet wanted me to put the $3 chick on a round of $400 medications, and she really couldn't even tell me what she thought was wrong with him, or if the medication was actually going to save or help him. When I chose not to, she charged me $60 to put the chick down...I left feeling sad for losing the chicken, sad for not having $400 for medication for the chick, and angry that I had just spend $155 to put a chick down when I could have just had my husband take care of it when he got home from work [but these were my first chickens and I loved them so much I couldn't bear to not take them to the vet first]
If you're going to transport, I'd just put it in a box with a towel, or wrap one of those hand warmers in a kitchen towel and put it in the box with the chick.
GL.