First, there is a distinction between not wanting the vaccine and not wanting vaccine mandates.
Greg Abbott supports the vaccine. He's against vaccine mandates. His primary opponent Allen West is against the vaccine, and currently hospitalized with Covid.
One reason for caring about whether migrants are vaccinated is that it seems hypocritical to have vaccine mandates on American citizens, but not on people coming from places where they have not been socially distanced, or tested for Covid, asking for help. For people skeptical of the vaccine, it's an indication that the left doesn't take it seriously (I know the situation is more complicated, but that's part of how it's seen.)
The United States is comfortable with holding immigrants to a different standard than citizens, one reason we allow for deportation after criminal offenses and try to ensure that immigrants don't become public charges. So someone who is against the mandate can be intellectually honest if they insist on this for people who are asking for aid. It's perfectly fine for help to be conditional.