They don't need him to say things. He can e-mail disability support if meeting face-to-face is too much. Just as he e-mailed his tutor.
But the needs of people with anxiety are subtle and diverse so we would not make any assumptions about what support would be appropriate without some contact with him. He isn't in a paradox he's in a double bind - if he were well enough to ask for support he probably wouldn't need it. He isn't reading Joseph Heller this term is he!?
So you are trying to break the double bind. He is able to talk to his GP. And he is able to talk to you. So all is not lost. You need someone inside the institution whom he can also talk to who can appreciate the scale of the situation and identify what supports can be put in place to help him. And as everyone said, the university will do a lot once they know what they are dealing with.
So the questions are a) how do you find that person and b) how do you convince your son to talk to that person.
It may take several goes, disability support services may be pro-active once you get through and be willing to approach your son after hearing from you. If they have a good mental health specialist person (we have 2 MH specialists) then this could be a plan.
The personal tutor is an unknown variable but you should keep pushing at that door - as others have said we don't get trained in this. MH happens to be part of my academic field so I tend to follow my students with interest, though I always refer them on to the appropriate specialists. Only disability support and/or the director of undergrad studies can coordinate the adjustments a seriously ill student needs. Has the personal tutor replied to the e-mail your son sent him/her last week yet?
The departmental secretary/most senior admin person responsible for the undergrad programme he is on could also be approached too.
If none of these people are responding helpfully I'd honestly try the Head of Dept just to get things moving. Explain that your son is suffering from severe anxiety, under the care of a GP and you are very worried about his mental state. You appreciate they cannot disclose personal info to you, but you do want to know if disability support services or the dept can make efforts to reach out to him and ensure that he is getting any support/reassurance he needs because he is acutely vulnerable right now.
I know if our HoS got an e-mail like that he'd be straight on it. We do have a duty of care to our students (and also we tend to like them and want them to be happy).
Letters from GP explaining the seriousness of the situation are helpful, but not essential. Any or all of these people should believe you and try to help. The real issue is not whether they should help but whether they can help, because to do that they have to be able to build up a relationship of trust with your son, which right now will be hard.
I do sympathise. It must be so painful trying to support him at arms-length like this. I hope that when the anxiety starts to calm down slightly and he feels less fragile he'll be able to see how worthy he is of help and support.