It's Possible...but will cost you $$$--Play the odds.
Yes it's possible to get the interview and hired (contingent upon the successful completion of your type rating) without many connections. In my interview group of 12 last April, I only know of 5 getting hired and three of us did not have our Type ratings yet. All three of us had a lot of IP/Evaluator/Check Airmen time, so my guess is that helped make the cut--I was not well connected at SWA.
From my personal experience, I had a little over 3,000 hours PIC Turbine Time, 1500+ IP/EP time with an ATP. The other military guy was over 2,000 hours PIC and the civilian guy was high time and a former Chief Pilot for a Regional. I'm not sure how they select who will interview, but from my limited observation...it seems like 1800 hours Turbine PIC and a Type or 2500+ Turbine PIC hours without a Type gets you in the "zone" where you may get a call for an interview (is that fuzzy enough language?).
As an interesting note, I'd only had my application in a couple weeks and had scheduled to get my Type in July at FTI in Denver (when I could get leave from the military), so was totally surprised to get the call for an interview so soon--be prepared as soon as you put the application in.
I was able to move the Type class up a couple of weeks, but it still cost me big seniority and $$ not having the type already (you have to be on property prior to Jan 1 for profit sharing). The other two people who interviewed with me in April started class beginning of August. I got my Type on June 20th and didn't start class until Jan! You get put in the pool on your interview date or type rating date--whichever happens last, so it makes a BIG difference in your seniority if you have to get the Type afterwards. Yes it was nice putting down the money for the Type knowing I had a job, but overall if you're serious about coming here, then I think it's worth playing the odds and making the investment up front (I'm not being a hypocrite...remember, I was scheduled to get my Type rating when I got the call).
Overall, I it's numbers and points driven by the quality of your app rather than who you know to get the interview. SWA is looking for Quality flight time, leadership, demonstration of excelling at past jobs, work ethic, positive attitude, and YOUR LOGBOOK. If you're high "quality" time then you may get the call without the Type. Otherwise, get the type. After you've interviewed, now the internal recommendations can help you. If you hadn't flown with many folks currently with SWA and are shy on the internals--it's not a nail in the coffin, you can still get hired.
Everything positive you've heard about SWA is true...it's worth stacking the deck in your favor, and it'll make a big difference in your seniority (shallow pool date) once you get the job.
Good Luck,
Speed