This kind of thing is a gamble. Most businesses fail within a few years, and a business that starts out without enough money or cash flow to pay their contractors would, in my estimation, have a higher-than usual rate of failure than one sufficiently funded to do things right from the beginning.
Also consider that their offer to exchange equity in the company is, in some ways, a bet they’re making against themselves. They’re making a decision to exchange a relatively small design fee for a potentially huge stake in a sucessful company that they could otherwise keep for themselves. Why would they even consider doing this if they felt confident of their own success?
Also, starting up a business involves all kinds of expenses: insurance, taxes, equipment, rent, licenses, attorneys, accountants, and the bills pile up from there. Are they making similar offers of equity to every vendor they need to buy something from? If so, they’ll soon give away the company and have nothing left for themselves.
If you were dealing with a solid company with a solid business plan, they’d have enough money to cover basic start-up expenses since the bank or an investor would have been confident enough in their business plan and chances of success to make the start-up loans. That didn’t happen, though or, maybe, your clients didn’t even bother to find out.
What you’ve described, points to amateurs on a shoestring budget, without a well-conceived business plan trying to get others, like you, to finance something they can’t find the cash to do themselves. From a purely statistical perspective, this whole thing will likely fail and you’ll be out whatever time you put into it.
On the other hand, I could be wrong, but if it were me, unless I felt strongly about their chances of success, I’d turn it down. For that matter, when working for start-ups having cash flow problems, I insist on cash in advance or I don’t do the work. Worthless equity in a currently worthless company, um, nope.