• If you've ever logged in to MySouthwest®@southwest.com and your browser had (and still has) form auto-complete enabled, your RR number is saved in the Account Number field.
• If you have searched for award travel without first logging in to MySouthwest®@southwest.com (with auto-complete), your RR number would be saved separately for the Rapid Rewards Account Number field on the "Retrieve your Award Information" screen. (Those login fields have different names in the HTML code, so they don't share the same saved form data.)
• If you have booked travel on southwest.com and supplied your RR number at time of purchase (with auto-complete), your RR number would be saved in the Rapid Rewards Account Number field that matches which passenger number you were on that booking (those fields have unique names so their saved form data is not shared with the login pages). If you didn't have auto-complete enabled but you received an Email confirmation, search your Email archives. I find the easiest way is to simply search for "ticketless" in the subject field.

If the self-service approaches don't pan out, you can always phone and request a lookup. Be prepared to answer some security questions to establish your identity. This phone-in advice is not based on experience, so YMMV.

The Add Rapid Rewards Number page is a great self-service way to add your account number to an existing reservation. You can also call southwest and ask them to do it for you.