Its a tricky one as my mum was registered disabled for years before she received any sort of benefit from it, beyond a bus pass. I'm not sure what "evidence" she had. As we saw from Voxi though, if you are on a low-income benefit, there is a mechanism now to allow a third party 24 hour access to your bank statement electronically to verify that. I went through that to get Voxi for £10/month for six months and it literally took a few minutes. But to verify being vulnerable but not eligible for a benefit for it, no idea.
On the flip side, the council do not seem to take our disabilities or health issues in general seriously at all.
Mum asked for the gas inspector to wear clean overalls (never been in anyone elses house except the wearer as she has allergic reactions to other peoples skin) and face mask during the inspection (were vulnerable). The contractor put this on their notes but the guy who turned up refused to honour them and the company claim they can't enforce it due to it not being legally required.