How do you say that you have been vaccinated in Latin? I'm not sure how to construct this tense, and I'm not familiar with a modern Latin verb for "vaccinate".
Vaccinate is already a Latinate word, so to go back into Latin is very easy. The -ate ending should indicate to you that the word is first conjugation:
vaccino, vaccinare, vaccinavi, vaccinatus
This makes etymological sense, because it's ultimately derived from a Latin word, vacca meaning "cow." The adjectival form of vacca is vaccinus, -a, -um (cf. bovinus from bos or porcinus from porcus). The word "vaccine" was coined because it was used to prevent smallpox by using cowpox, the Latin name of which was at the time variolae vaccinae.
Grammatically, this is straightforward. To express a first person singular ("I") perfect passive ("am vaccinated") verb, you simply use the fourth principle part with the present tense of esse. This gives you vaccinatus or vaccinata sum. The former is masculine, the latter is feminine.
You'll want to use the perfect tense here, because the action of being vaccinated was wholly completed in the past. By saying "I am vaccinated," you're really saying, "I have (already) been vaccinated." You're describing the present state by means of a singular past action. You can see the same effect in very beginning of the Caesar's Bellum Gallicum: Gallia est omnis divisa in partes tres, "All of Gaul is divided (='has been divided') into three parts."