Lack of infrastructure isn't the impediment - the issue is simply that the Tories are reluctant to go down that path. Presumably they think their incompetence hasn't killed enough people already, and would like to take a few thousand more of us out before this is over.
There's a vaccine passport system ready to go - at least for anyone with a smartphone, which these days is most people - because there's a vaccination status page on the NHS app. I don't know whether the law would preclude a producer from unilaterally requiring patrons to show proof of vaccination status or a negative covid test. The issue is simply that the Conservatives, who brought us an even higher per-capita death toll than Trump gave the US, have their heads planted firmly up their backsides - and since they have an 80-seat majority, there's not a lot the opposition can do about it.
(No I didn't vote for this government. Does it show?)