IMO, for their safety, the cat must remain in their carrier.
If a car accident were to occur, the cat would be safer contained in the carrier than loose in the car. In addition, if the accident broke windows, the cat could be ejected and/or leave the car in a panic on its' own and be hit or lost. Keeping the cat in the carrier also prevents the cat from panicking around the vehicle interior, distracting/injuring the driver, injuring passengers, getting lodged under critical control pedals or wedged under seats. Also, any cat accidents (vomitting and voiding of bowels or bladder) is contained in the carrier. I always put a thick towel in the carriers. Shasta always burrows under the towel and hides when she travels in the carrier.
I try to position the carrier with the door pointed towards me so I can see them and give them eye-blinkies. I will sometimes push my fingers through the openings for them to rub their faces on. If it is a hot day, I will point AC vents to the carrier vent openings to help the cat keep from over-heating. If the cat is a very nervous traveler I will cover the carrier with a towel so they don't see everything wooshing past.
My vet is only 3mi from my home.
1mi of 55mph rural road. 1mi of state freeway, 55mph. 1mi of in-town 35mph road with only four street lights. This takes well under 10min (5-
To keep the drive as stress-free as possible:
Once the cat is in the carrier I get immediately into the car and go. I do not wait around to gather keys, handbags, wallets or whatever. All of that is done and placed in the car before the cat goes in the carrier. I drive directly to the vet and directly home afterwards. No stops, no matter how 'quick' it would be. I will take the cat home and drive back to town to get whatever was needed in a seperate trip so the cat can get home and de-stress as soon as possible.
Reilly was the only cat we've ever had that LIKED to ride in cars. The rest, just have to deal with it until the ride is over. I do what I can to minimize the stress, but it is a necessary evil that they have to ride in the carrier/car to get to the vet.
Yearly shots are done in my home by my horse vet when she comes to do routine shots for my horse. One ranch call of $70 covers her coming to treat my horse and my cats at my property. I've even split the ranch fee with a neighbor when she brought her dogs over on the same date (scheduled together) for their yearly and rabies shots.