He dislikes the vet because he's not had a very nice time whenever the vet is there. He'll be like that with every vet (male and female) unless you teach him not to be - vets don't tend to come to see your horse for social calls, vet visits more often than not involve being poked, prodded, jabbed or some other painful or unpleasant thing. And your horse can smell a vet at 100 paces (honestly, they don't smell bad to us, but they do smell distinctive

).
You don't need the vet to practice. You just teach your horse a routine - for mine it's running my fingers down either side of his jugular - which is always followed by him relaxing his neck and me stabbing him (fairly gently

) with a sharp object and him then getting a reward. When the vet arrives, you tell them the routine - i.e. approach horse from left, stroke fingers down neck until neck relaxes, prep injection site, stab. Provided the horse knows exactly what happens next and that it has never been scary when you did it in rehearsal, they will be fine.
There is no easier way to do it, unless you want to sedate your horse every time the vet calls. And to sedate your horse effectively, you need to inject them. It's easier for the horse in terms of welfare to prepare them for nasty things that might happen in their life and to teach them that nothing is
that bad - and it's safer for the vet if the owner has taken the time to train the horse.