@MrsDoctorDear - yes, GPs are doing visits to give the vaccine to patients who are truly housebound. You can’t book any appointment with a GP-run Covid service via that booking link though - the link is just for pharmacy-led sites and the big Vaccination Centres. GPs are doing all their own invites using their own patient lists.
If your DM is currently eligible, ie she is over 70 or clinically extremely vulnerable (being disabled isn’t necessarily clinically extremely vulnerable), then her GP should be contacting her imminently, or sir could give them a call if she thinks she needs a home visit.
If she’s not yet eligible, then her GP will be in touch when she becomes eligible.