No. So many reasons.
* Even people with long-term commitments to companies have a hard time valuing equity.
* In order to value equity, you need to be given access to confidential details of the company.
* It's legally expensive to give different kinds of equity to people, and every time you do it you create a small (or worse) amount of risk.
* The rules about employee equity are well-tested and understood. The rules about equity offered like this aren't.
* Offers for exchange of equity for in-kind services could be construed as unlawful solicitation of investment (I don't know, and am not a lawyer).
* Having a web design contractor on your cap table is going to make it harder to close VC rounds.
* Screwing up your equity grant to a web design contractor so that they have an effective veto on a VC round is going to make it impossible to close VC rounds.
* Employee equity vests.
* At good companies, a grant of equity has uncapped upside. Nobody buys web design for "potentially unlimited dollars".
* So now you also have adverse selection to deal with: the companies whose equity is available in a program like this won't be the Airbnbs and Dropboxes of the world.
* Similar barter programs (based on pure in-kind/in-kind exchanges) have been tried for decades (the ISP I worked at in the '90s was involved in one) and they appear to reliably fail; once people start to believe their contributions aren't fairly valued, a vicious cycle sets in.
I wouldn't just not participate in a program like this; I wouldn't work for a company I found to be participating in one.