Grammar – Is the Sentence ‘Did You Get Vaccinated?’ Correct?

Speakers of American English are more likely to say

Did you have a chance to...? OR
Did you get a chance to...? OR
Have you had a chance to...? OR
Have you gotten a chance to...?

I cannot say whether this is also the case among speakers of British English, who would say the last as Have you got a chance to...?

All have the same significance: they all ask whether an opportunity occurred. Consequently, answering in the negative does not mean that there was an opportunity but it was missed: it means that the opportunity was not there.

Generally speaking, only the first verb in each clause is finite:

They knew what it meant.

Here, know is the finite verb. It changes form to agree with the subject (I know, she knows) and to indicate tense (They know, they knew).

To turn this into a question, we need to apply Subject-Auxiliary Inversion. But to do that, we need an auxiliary, so we add the dummy auxiliary do:

They did know what it meant.

Did they know what it meant?　　←　did and they are inverted

Now do is the finite verb. It changes form to agree with the subject (I do, she does) and to indicate tense (They do, they did). The second verb, know, is non-finite and does not change form.

*Did they knew has two finite verbs and is ungrammatical.