Have a large and mixed population, release a lab created virus that is strong enough to kill really old, sick or otherwise unhealthy people but simply make everyone else sick for a little while.
Tell the entire Vault that they accidentally released said virus and require everyone to wear masks and to stand six feet apart from each other at all times while they work on a vaccine. Don't specify when if asked, just keep telling them you're working on it.
Have everyone follow strict individual curfews so that only certain amounts and kinds of people are allowed to go out and do what they need to do at a time. Any physical contact is punished with solitary confinement for two weeks for both the person who touched and the touched person.
All the while, keep playing a PSA over the Vault intercom to stay six feet apart, wear a mask, wash your hands often and to eliminate physical contact every five to ten minutes.
When the vaccine is released, release just enough so that at least half of the population gets vaccinated, but withhold the rest and tell the rest of the Vault that they'll get their vaccine in short time. In addition, alter a small amount of doses of the vaccine so that certain vaccinated individuals suffer extremely debilitating pain, horrifying side effects and death. Induce paranoia and fear into both vaccinated and unvaccinated.
Allow vaccinated free access throughout the Vault. Give them no curfews and allow them to do as they please while keeping the rest of the population under lockdown as well as wearing masks. Start spreading propaganda about how deadly the virus is and how you should be vaccinated if you haven't already. Guilt and shame those who have yet to be vaccinated despite not being able to get it. Meanwhile, treat the vaccinated as if they are heroes and good citizens, telling them that what they're doing is saving lives.
Release new and more contagious variant of the virus onto the population that affects the vaccinated after a while. Force vaccinated back into wearing masks and curfews and tell them you will make a new vaccine for that. Spend a large amount of time making new vaccine and release it to the unvaccinated half of the population. Give them the same privileges as the other half did before getting afflicted and praise them for being heroes while shaming the other half for not getting the vaccine.
Rinse and repeat.