Yes and no. The airline pilots are unionized but in their own separate union from the rest of the pilots. The flight attendants are also unionized. It’s not clear if they too are separate from the other airlines.
Southwest has often been shunned by other major carriers. That’s why you do not see them listed on places like Orbitz and Expedia because the major unions tried to get them to join and Southwest refused.
My point is that the union doesn’t wag the dog at Southwest like it does at American, United, Continental, etc.
You don't suppose the reason why Southwest was shunned by Orbitz might be that it was founded by Continental, Delta, Northwest, United and American do you?
Well...after twenty+ years in the airline cockpits (nonunion, union, independent union, ALPA union), this has not been my experience. American has an independent pilot union, they are listed with the companies you mentioned. US Airways is an independent pilot union, they are listed as well. You might consider that pilot unions have virtually zero influence on business arrangements such as this. I fear you give unions too much credit. Unions simply don't care about these arrangements as it is of no consequence to them, and outside of their contractual capabilities in any case.
Perhaps I can clear things up a bit. Consider, if you will, why Southwest would not choose to associate with these web travel companies. Could it be as simple as Southwest not wanting to pay the commissions associated with having Expedia selling their tickets for them? This cuts into their bottom line. Could it be that Southwest is perfectly capable of handling their own reservations, thus preserving their profit margins? Southwest does seem to have a different type of customer. At least consider this as a possibility and do your own research on it of course.
I am not hammering you. It's just something that you should consider as an explanation for these business arrangements (assuming that I am correct on the limitations of unions in such matters).