Ok, well first of all, I don't think action needs to be taken against the volunteers. My god, the voluntary sector already struggles to attract and keep volunteers as it is and with budgets being cut to the bone, it's only going to get worse.
You are absolutely allowed to ask for information, even sensitive information if you have a purpose for doing it and you treat it appropriately. They obviously think they do. That might be flawed but the fault is with the organisation. They ask the same information when taking a booking so there's obviously some process they think benefits from this information. They definitely need to be more consistent with their volunteer training so the volunteers know what they should/shouldn't be asking and why.
The chairman's attitude is the biggest problem. He should be able to articulate why they take that information and what they do with it without being rude or defensive but one rude interaction isn't enough to push the nuclear button.
A lot of small organisations are doing the best they can under a lot of pressure because statutory services are shrinking and leaving huge gaps to fill. I don't think they should be completely let off with bad practice but a bit of flexibility is needed.
If you really want to take action, ask for their privacy notice, you could even ask for a subject access request too. It should lay out what data they capture, why and what they do with it. If you're not happy with the contents or they refuse to give it to you, you could access the organisations information through the Charity Commission and write something to the board who should be identified there.
I wouldn't start making waves elsewhere until I'd done this, at least.
By the way, I'm not even suggesting OP is even bothered enough to go this far. I just think it's a better route than is being suggested here. At the very least, I'd use the information from the Charity Commission to address a complaint about the chairman's conduct to the wider board.