That is not correct.
You can object to receiving any or all vaccines if you have medical conditions, religious, and/or moral beliefs. You just have to be sincere about it and attach a waiver form. It is somewhat hard to prove since the reasoning may be very subjective but there’s an option.
Read the FAQ from the link that Misomasr attached to his reply. There’s a question about that.
Q. What will happen if I refuse to receive one or all of the required vaccines?
A. Tell the civil surgeon if you do not wish to receive the required vaccines or a particular vaccine. You should also tell the civil surgeon the reason you do not wish to receive the vaccine(s). In this case, a waiver may be available to you, but only under the following circumstances:
- You are opposed to vaccinations in any form– that is, you cannot obtain a waiver based on an objection only as to one vaccination
- Your objection must be based on religious beliefs or moral convictions; and
- The religious or moral beliefs must be sincere.
The form used to apply for a waiver depends on the adjustment category under which you are seeking legal permanent residence status. For example, refugees and asylees seeking adjustment of status should file Form I-602, Application by Refugee for Waiver of Grounds of Excludability. Individuals seeking adjustment of status as a result of an approved Form I-130, Petition for Alien Relative, or Form I-140, Immigrant Petition for Alien Worker, would file Form I-601, Application for Waiver of Grounds of Inadmissibility.