I think "anti-vax" sounds inherently extremist and conspiracy-theorist, and isn't an accurate representation of the middle ground here. To be fair, it's more like "pro-choice."
Plenty of vaccines are given to children every year and required for them to attend school, and there hasn't really been any negative side-effects of that policy. It's tough to argue that we should not be required to have any vaccine to attend school, and that would break the status quo in the other direction.
The question then becomes: What could make the Covid vaccine different from other vaccines, such that it shouldn't be required?
* If these vaccines were approved earlier and with more early-stage preliminary data than other vaccines?
* If Covid is not as severe as the diseases we currently require a vaccine for? [1]
[1] The current required vaccines for school in New York, for example, are Diphtheria, Hepatitis B, Measles, Polio, and Chickenpox. The mortality rate and permanent damage for some of these diseases are much higher than Covid. With others, I'm not sure of the pre-vaccine mortality rate as it's been around for a long time.
With all that in mind, Covid spreads as much if not more as the diseases we saw fit to require vaccinations against, and the mortality rate for the older population is high. I think a fair argument is whether schoolchildren should have to be vaccinated to protect against infecting the more vulnerable population.