We have been going to BC, Canada just about every year with 2 - 3 dogs for a long time now. Entering BC, they require current rabies shots, so we carry their shot records. Most of the time they just ask and don't look at the records. It's useful to have, too, in case your dog gets sick and needs to see a vet.
Entering the US, they want to know if the dogs were born in the US but don't ask for documents. Although, if asked, the shot records would be enough proof. I guess if you were importing a dog from Canada you would need papers. The other thing they sometimes ask about is the dog food you are carrying. There are lists from the Dept of Agriculture that limit certain meat products entering the US (even in dry dog food). We try to bring enough dog food to last through the trip. If we need to buy some in Canada, we buy small amounts so there is very little left when we come home. In case, they want to confiscate it but they never have.