A better outcome would increase the NHS rollout, so more people benefit. A (different) successful petition on the above government platform could help with that.
If the OP is correct in suggesting that selling the vaccine is already legal and it’s just a matter of consumer demand, then I don’t see how petitioning the government will help.
Is it now possible to sign up?
This petition may be a substitute for the OP plan: https://petition.parliament.uk/petitions/623325
A better outcome would increase the NHS rollout, so more people benefit. A (different) successful petition on the above government platform could help with that.
If the OP is correct in suggesting that selling the vaccine is already legal and it’s just a matter of consumer demand, then I don’t see how petitioning the government will help.