Of course that becomes a problem is a large part of society does the same. Which is why such laws against the exists. But I wouldn't be the one pursuing this.
Especially for online spaces. There's just too many options for it to be reasonable to be pissed about getting banned from just one or a few places. It's always possible to create your own online space, if necessary. And right to speech isn't a right to an audience. You get to speak your mind, but people will only listen if they want to.
But keep in mind that there's a bet good reason for why most antidiscrimination laws deliberately limit the contexts in which they're applicable and what groups they protect. Very few jurisdictions protect opinions on their own.
I don't care if an antivaxxers works at a bakery. I'd be pissed if I hear one worked as a nurse at a hospital I go to, and that they refuse to carry out vaccinations. Some opinions are dangerous when held in certain contexts, and requires a reaction. This is simply about safety and protecting society. You can hold an opinion that terrible opinion elsewhere, where you don't hurt people.
At some point, you have to understand that the person putting their livelihood at risk is the person themselves, and that firing them is just as justified as firing a drunkard from a machine shop.