Huey-I disagree. Check out this part from the webpage:
"1 A Candidate may apply without a B-737 Type Rating. If a candidate interviews and successfully completes the entire selection process, he/she has six months from the date to obtain a B-737 Type Rating to be eligible for hire."
Link--http://www.southwest.com/html/about-southwest/careers/positions/pilots.html
I have no type rating either, but if I am lucky enough to get that far, you can dang sure bet I'm selling a few teeth to add it. good luck.