Recruiters tend to insist on a résumé because it allows them anonymise and commoditise the resources they're selling to their clients.
By sending them a résumé you're playing their game and probably selling yourself short.
An actual client might ask for a résumé as well because she's used to receiving one but she'll probably be as happy - if not happier - with a website that not only details what you've been working on and how that benefitted your clients (rather than a cryptic bag of TLAs) but maybe also provides client testimonials underlining your ability to deliver.