For sure...
The vaccines are provided by PharmaDoctor.
https://pharmadoctor.co.uk/
They do health checks, you can choose your vaccine (subject to availability) and it's done at chosen pharmacies, and (for me, quite important) they inform your Doctor so it shows on my NHS app.
I know about this because Jenny Eclair (similar age to me!) wanted one and asked on Twitter and a Doctor replied, so I went for it!
If you want to know anything else, just shout 😂