Well, at least they've managed to put up a banner confirming that they're having issues, instead of us just getting random errors:
Book a Flight is undergoing maintenance. We are experiencing flight booking issues due to a global vendor disruption. We are working with the vendor to restore this feature as soon as possible. All other site functions are working normally.
If your flight booking is urgent, please contact a Southwest® Representative at 1-800-I-FLY-SWA® (1-800-435-9792). There may be high call volumes so please wait to call if your trip is not within the next 72 hours.
Thank you for your patience.