Update: Literally minutes after posting this, the announcement came that he got the jab. So never mind.
If you were managing the Warriors, how would you deal with a player who is doing what Wiggins is doing?
Whatever you do, you should make every allowance for the situation to resolve itself in the easiest way possible, for as long as you can. The simplest, easiest solution is for Wiggins to come to his senses and get the shot, obviously. Everyone could then congratulate him for considering the factors and making a sound decision. His teammates could express their support and satisfaction that he'll be there with them. And then the whole, united squad could go get after it.
That's the best case scenario, so whatever the team does should be done with that outcome as the goal. No threats, no shaming, no talk of hasty trades. Just give him all the space he needs to make the obvious choice for himself, which in fact is a pretty good description of how they've been handling it so far. Don't panic. Don't say anything you can't take back. It's hard to be patient in the face of such abject stupidity, but that's where we're at. We might even have to keep being patient into the beginning of the season. There's no way to know what it's going to take.
If this does go on well past the start of the season, however, the team should look after its own interests. If a player refuses to make himself available, the team has every right to minimize the financial damage it causes them. They can withhold salary for the home games he'll miss. It's a lot of money and a lot of games to miss. But what about road games? Apparently they have to pay him for those. But should he play? An argument can be made that the team might be better off benching him. It'll be disruptive to the lineups and the substitution patterns if his availability is so sporadic. Having him available on the road might be worth the disruption it will cause or it might not. I'm not sure. Another factor is financial risk. If he plays road games and gets hurt, that would mean he'd then get paid for the home games he misses--you can't withhold compensation from a player who's not available due to a basketball-related injury. So Wiggins would be financially incentivized to get hurt during a road game, which would be an absurd situation. For these reasons, I say bench him on the road.
Also, if it can be done legally, bar him from the team facility for health reasons, and fine him for missed practices. And require him to travel for road games (that he won't play in) but not on the team plane, also for health reasons. Interestingly, there's a bill in the senate that would require proof of vaccination for domestic flights. If it passes and he can't fly either with the team or commercial, he'll have a tough time getting to games just to ride the pine. It might enable the team to not pay him for road games, too. This one definitely crosses the line into the area of punitive measures, so it runs the risk of alienating and angering other players and thus should be considered carefully.
For now it appears the Warriors and the Nets are the only teams that have to deal with this. It's probably just a matter of time until other cities pass similar ordinances. 95% of players are vaccinated, reportedly. It's unclear how big of an issue this will become league-wide. Until it becomes a bigger problem for more teams, I would expect the NBA to continue to basically do as little as possible and say they lack the authority to mandate player vaccination. So the Dubs (and Nets) have to look out for themselves.