Despite what the web page says, SWA will rarely even interview anyone who does not have the 737 type. I guess they have enough applicants willing to pay for it or who got one somewhere else...
Essentially all regionals, LCCs (except SWA), and majors in the US do not require you to buy aircraft-specific training. Some airlines do have a training contract to ensure you don't take their training and then quit in 3 months to go work elsewhere. There are also a few "pay for training" scams out there where very low-time pilots pay an airline or their affiliate flight school a large sum in exchange for training and a job.