Is this an elderly dog? Have you considered one of those dog back packs or front carrying packs instead? Much more easy to pack in your car!
Also most national parks, if this is where you are going, will want you to have your dog on a leash anyway. Bells would be good to put on the collar if they are walking so as to warn the bears that you are coming if you go on a nature walk. Other wise you put bells on you somewhere
Be aware, be safe
Some places you won't be able to take a stroller anway.
Hope you all have a great time!