You will all have heard about the dog that HATED the vets, knew they were going to the vets and 'acted up'.
This is a very common finding - most vets do not read dog body language well and, in order to do their job, need to handle the dog to figure out a problem or issue the dog is having.
If your dog cannot walk on a lead we advise only going to the vet if it is absolutely necessary, and then they must be crated with the vets being told prior to arrival that the dog cannot be touched. It would be better if you could request a video consultation for nervous dogs.
It is entirely possible to build up a positive relationship with your vet and have your dog comfortable when meeting the vet team, but there will be work involved.
First, ensure your dog is comfortable with handling, there's no point in forcing them to be touched by a stranger if they are not going to cope with it. All that will happen is the vet will make the dog wear a muzzle without training and make it very frightening for the dog.