To be honest I don't think there's much time to start mucking about with hypnotherapy. I am sympathetic to phobias I'm not just being callous here. He's a grown man, he's been offered a vaccine that people are absolutely desperate for. If you really must, ask the GP for some diazepam but I don't know that they'd really do that.
I'm afraid whilst I'm sure I will get shouted down because .. phobias are real..I get that though, but he just needs to go. Someone needs to go with him and he needs to be as adult about this as possible. We are in a pandemic that we are all sodding fed up with and has been going on for nearly a year. The only thing that's going to get us out of this is vaccinating people who can and need to be vaccinated. I assume if your brother has been called he is vulnerable.
I've been called to have it - I can't, I have very severe allergies. There are others like me, there are CEV children by the bucket load who can't be vaccinated as it isn't for children - and they, and all the adults who can't have it , rely on everyone else having it for their protection.
It's your brothers choice whether he has it or not, and my above point I've only made because you said he does want it. If he wants it I'm afraid he is going to need to walk in there and let someone stick a two seconds needle that isn't going to hurt, into his arm.
I'm sorry for being blunt, but he wants it, and the more people who have it, the less the risk of serious illness and hospitalisations is, and this ends quicker.
I'm not going to call him silly for having a phobia, but unfortunately my only answer is just go and get it done.