Here's what I ultimately found, and I'll address a couple of points that xochi mentioned.
SendMail does work in WebDirect, with two limitations. You cannot use the "Perform without dialog" option and you cannot attach files.
If you choose the "Client Email" option, it doesn't do the mailto thing (You can get it to do that using the OpenURL script step, though). It does open Mail (on Mac) or Outlook (Windows) or whatever e-mail app is present on the device.
My main problem was that I had pop-ups blocked in my web browser. This was what was preventing the email client from opening. The browser considers that a pop-up window. It had nothing to do with FileMaker, WebDirect, or even the device. It was a web browser setting.
Once the pop-up setting in the browser was changed to allow them, SendMail did work as expected.