Should I ask for equity when joining a startup? When/how? (And whom should I call via Clarity?)

If the team has aspirations to build a venture backeable business (i.e. Raise money for equity), then of course you should ask. Ask the CEO if he has a ESOP (Employee Stock Option Plan), and if it's part of the compensation package.

You're allowed to ask.

As for the right person on Clarity, pretty much anyone in this list
https://clarity.fm/browse/raising-capital/venture-capital

Tom seems to be fast to respond.

Not only should you ask, but you must ask. You may not know it, but your personal credibility is on the line. If you are worth "x" on the market and you ask for substantially less than "x" (or you fail to ask at all), then you are communicating the message: I will not stand up for myself (which, in turn, suggests you will not stand up for what the company deserves or to a vendor or employee against whom a stand needs to be taken). You must show courage, confidence and conviction because if you can't make the right deal for yourself, then you won't be trusted to make it for others.

False humility is as harmful as hubris. Seeking what you deserve based on your skills is not about "greed"; it's about value. So tell them what you deserve, tell them you understand that accelerator covenants have to be followed, and then say something like: "I understand your current restrictions on equity today, and I understand you're cash short. I'm very interested in this, but you're asking me to take all the risk (no equity, no cash) without any assurance of reward even if, as I expect, I exceed your expectations. If you were in my situation, you'd have exactly the same concerns. What do you propose to balance the risk and to ensure I'm compensated fairly once this gets off the ground?" Then don't say anything, even if you're silent for a minute or two. The onus is on them to address your concern.

Of course, this is best done in person or at least via skype. This is a discussion to have with the CEO and the sponsoring mentor/investor in the accelerator, not with others. Remember, they want you. Remember, they already have equity and you don't -- and don't be afraid to bring that up, not defensively but as a point of fact. Even in the worst case -- i.e., there is truly nothing in the world they can do (which I doubt), you'll be respected for having brought this up and you'll have substantially increased your chances of getting a good deal later. Good luck.

You of course should ask for a compensation that is appropriate to the qualification and your role in the company. In most of the cases a startup can not financially compensate people at the right level - that's why they are offering options and equity to close the gap. The only other way is a deferred payment (they are saying that they will pay you back when they get funding). But equity is the most common way.

Talk to founder(s), CEO and CFO...
There is no 'rule' on how it is done, but the earlier you join, the bigger piece of the action you should be able to claim.