Yes a small cat or dog carry can work, a lot of people do that, just line the bottom with fleece, and put some hay in there to munch on. If possible one that isn't all plastic would be better, but so long as the drive isn't hours, and your kit isn't like my two when I brought them home it should be fine. Most chins just sleep or are too freaked out to really do much in the carrier so it's not normally a problem.
I got my current two from the next town over, about a 45 minute drive, and in that time they had chewed holes in the cardboard box they were put in and by the time we got home I had my hands over the holes to prevent them from getting out. Luckily I wasn't the one driving, and they came with their cage so I we were able to quickly set it up and put them in. We had planned to get a pair of chinchillas but didn't plan to get chinchillas that day, we went to the feed shop for cat and dog food and they were being sold by the owner for someone that moved and couldn't take them with them, the shop didn't normally sell chinchillas.
What I currently use as their travel cage/carrier is this (it also doubles as their bath house cage).
View attachment 21322
(old pic I have a different bath now)
It's just a plastic base but I can just put a piece of fleece over it to prevent chewing. You can get smaller size ones too, and even a small one with a handle.
https://www.amazon.com/dp/B000MD3MH8/ref=twister_B07K8CQMVB?_encoding=UTF8&psc=1