If you get a secure carrier of a decent size and fill it with soft bedding and a little bit of food plus a chunk of cucumber you hamster will travel well I'm sure. They normally sleep once settled. Try and keep it dark so maybe cover with a towel or blanket abd don't play loud music in the car on the way. Most hammies do travel well. Am alternative is a plastic storage box about 1ft square with holes drilled in the lid for air. I hope you don't need to take you hammy to the vets though.
A guinea pic carrier like this is good
https://www.amazon.co.uk/Ferplast-Al.../dp/B00F5ZYR72