Apparently, this DUmmy doesn't know that being up to date on all vaccinations is a license requirement for the majority of medical providers. My wife is a nurse and not only is she required to get every seasonal vaccination for things like the flu and COVID, she also gets a blood draw once a year to check antibody levels from previous vaccinations. A few years ago, she had to get an MMR booster because something popped up in her lab results. There aren't any licensed American doctors or nurses who aren't up to date on their vaccines and none of them would just randomly bring up the issue. But maybe LuckyCharms was seeing a doctor in Mexico. Or maybe when he says "doctor appointment" he means trip to local marijuana shop. And maybe when he says "nurse" he means cashier at the local marijuana shop. Because he does spend a lot of time talking about getting stoned on whatever weed or gummies he got his hands on, and they do like to bring up their belief that marijuana is the ultimate medicinal marvel.

for the info.
It may also be that the "doctor" in this story is its shrink (psychiatrists are MDs), and the nurse, knowing
ChuckyLarms' socio-political views, trolled it with a fauxny

just to see if it bought it.
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"Anti-vaxxers" are a mixed bag. Some really believe vaccines cause autism or other nonsense and eschew vaccinations entirely. Some, as you said, oppose government requiring vaccinations and/or the aggressive schedule of the past few decades. Those who oppose vaccines entirely are also a mixed bag when it comes to political views, some conservatives, some Libs/Progs
(a conservative family acquaintance/friend literally believes that Moderna has hidden hundreds of thousand of bodies of people killed by The Vaccine; OTOH, on a now-defunct homeschooling discussion site, I saw anti-vax posts from anti-Wester-Medicine Libs).