First, illegal immigration is illegal so you have to catch the person to vaccinate them. Second, we don't force anyone to get the vaccine that's already here in the country so we aren't going to force an illegal immigrant to be vaccinated just because they were caught. Just like if an unvaccinated person is arrested for shop lifting we don't vaccinate them in the squad car against their will. If we aren't going to put them through the system to send them back to their home country and they want to stay then I would mandate vaccination.
Third, an asylum seeker isn't an illegal immigrant if they go through the process so these people, I believe, should be getting vaccinated to legally come into the US.