Dog afraid of vet

Yes, your vet is your go-to person here, they're used to it and its not nice for them either.

My dog use to shake if we drove down a street that was 3 streets away from the vet. It's very sad to watch.Can you phone the vet to ask him if he can give your poor little pooch something?

I have had this problem with my dog and his brother who died last year. I would say be careful because my boy who died, while lovely with everybody else, and lovely with vets in his own home, got really dangerous at the vet when he had bad ears and a vet almost got bitten. Luckily they know me well and we came up with an action plan to continue the needed treatment safely. It involved teaching my boy to be muzzled and for me to do most of the handling. ALL of the practice were lovely with them right up until the day he was PTS in my garden.
there are things you can do but they involve a lot of work. Good article here but in your case it wouldn't begin with the vet but with the street where the vet's office is. Whatever you do please don't (and I don't think you would) take any advice from anyone that the thing to do is to force your dog to face their fears... I mean if you were phobic about spiders, how would you feel if you were FORCED to hold one? and its worse for dogs because they can't understand what is happening.
I actually took a different route because I was in a different position. I decided to cut out all uncecessary vet visits. The dogs stopped vaccinations and check ups and I arrange home visits for necessary stuff a far as possible....mind you my dogs are bigger so harder to handle against their will....I can't just pick my remaining boy up for instance. I discussed this with my vets and they are fine with it and very understanding...I think they like that I am concerned about their safety.
link thedogtrainingsecret.com/blog/understanding-desensitization-dog-training/

You can get something called Calmex from the vet. We use to give our previous border collie, Charley a couple an hour before his appointment. They are not on prescription so can pick them up at the desk.
They relax them and he wasnt stressed out when he needed to go.
Unfortunately, Finn, our present BC, has the opposite problem, once he is inside the vets, he stages a sit down and refuses to leave. Recently it took ten minutes of persuasion and kisses to get he outside.

I had similar problems with Gracie- it once took 2 vet nurses, a vet plus me to get her up the ramp to the door and inside!

I have a theory that dogs can smell fear or illness from where other dogs have marked(wee’d) outside the door or in the carpark - like a warning to other dogs.

We got round it after a fashion by not letting Gracie have a sniff or wee on the way in- I would just run her up the path and in before she realised what was happening! Or perhaps a vet nurse could collect her from you a few yards away, in the car park if there is one.
It is heartbreaking to see, but usually over quickly and lots of cuddles, fussing and treats may help to remove the memory. Just to add , when it came to the final goodbye, our lovely vet plus nurse came out to the house so that Gracie went to sleep on her own bed.
Good luck!