Your experience and actions deploying for Desert Storm may have been a valid tactic back then, when the Air Force less digitized and hundreds of thousands of people/dozens of squadrons bigger. It isn't any longer.
In my corner of the AF (can't speak for others), your immunization status is tracked via ASIMS and reported to Sq Mobility and leadership every week. If you're overdue a shot, you get told to get it politely, once. Then you're in the DO's office explaining why you're red on immunizations and therefore not ready for alerts or deployments. Putting aside this recent COVID dicknannagans since this is supposedly about monkeypox, getting vaccines if you're a military member isn't a choice or negotiable based on your personal feelings; it's a condition of employment.
I didn't particularly want to get Yellow Fever, JEV, Anthrax, Smallpox, or Rabies vaccines, but they were all required for me to deploy, they're required for me to continue serving and a lawful order, so I rolled up my sleeve and got them. Much like PT tests, OPRs, or ancillary training, I may personally think some of them are stupid or silly but the corporation didn't ask me and doesn't give a shit about my opinion. If I or anyone in my squadron starts to refuse vaccines, they're now non-deployable and a drag on the squadron cause we've got to find someone else to cover alerts or deployments.
If an individual's personal feelings/beliefs/analysis about vaccines are so strong that they refuse them, then it's time for that person to find different pastures besides military service.