Which other businesses? People will simply buy forged records I would think.
Really.... it seems that absolutely EVERY effort to do anything about coronavirus has suffered attempts to shut it down on the grounds that "Well, people won't comply/won't stand for it/they'll lie and break the rules." And it's usually turned out to be wrong.
Travel bans were disputed on teh grounds that "people will find ways round them" (actually, travel bans mostly work once they are in place. Few people can afford a private jet, and most people are spooked by the thought of being refused at the airport somewhere and left completely stuck)
Lockdowns were disputed on the grounds that "people will refuse to comply" (actually, people mostly did comply)
Facemasks were argued about on the grounds that "some people have to be medically exempt" "people will lie and find reasons why they can't wear them" Some people have but most people have complied.
etc. etc. etc.
Most rules do have "ways around" if a person is prepared to spend a lot of time and energy finding one, but the reality is that most people will comply if a rule is in place, and it's better to have 99% compliance than 0% compliance.
Also, small shops are another matter, but for things like international travel, event attendance and other big-ticket entrance points where you usually pre-book and pay in advance, it will be easy enough for the officials in charge to CHECK against centralized records of vaccination. The vast majority of people would not do things which might create the risk of being hauled in for questioning by airport security people or immigration officials.