Solution: All SWC join requests need to be manually approved by an available admin, so this can take some time. However, this will generally not take more than 12-24 hours max, so if you have waited a full day, chances are your account was already approved or rejected but you did not receive the e-mail for it.
If this is the case, be sure to check any spam folders in your e-mail in case the response was blocked by a spam filter. If that turns up nothing, go to http://www.swcombine.com, click the Lost Password link, then enter your e-mail address into the textbox.
If you receive an "invalid e-mail" error, it means your application was rejected, and you can try to sign up again with that e-mail address. If you receive a "Successful!" message, your password will be sent to you a second time.