We're flying Southwest. Only one of us has a cellphone but I'm not sure she has the ability to get onto the internet.
First thing to know is that it's the checking-in part, not the printing-out part, that's important. As long as you get checked in, you can print out your boarding passes when you check in with RAC (if you do that) or even when you get to the airport. You'll still have the same numbers assigned at check in no matter when you actually print out the boarding passes
To check in for your flight with SWA from WDW you have several options. The first and most convenient is to purchase their Early Bird Check-In for $10 per person. That not only gets you checked in 12 hours before you could do it yourself, but it also means you get checked in automatically and don't have to worry about it during your vacation. You can access your reservation on southwest.com and add EBCI.
If you don't want to pay for EBCI and don't mind taking a few minutes out of your day, the guest services locations at any of the parks or in DTD or the front desk at any WDW resort will do the check in for you. You will need your confirmation number(s) and also may be asked for a photo ID.
As noted earlier, you also can check in online from any Web-enabled cell phone or other device (no app necessary) at mobile.southwest.com. You will need to enter your confirmation number(s) and one passenger name per confirmation number.
Finally, you can give your confirmation number(s) and associated passenger name(s) to a trusted (100 percent reliable) friend or relative and have them check you in from their computer.