I suggest waiting till about one day before your departure date. Technically, you could wait till ten minutes before departure, but I would not do that. Practically speaking, you are still okay, if you want to cancel up till about two hours, before departure; but, just in case there are any issues, I would not go later, than that. If Southwest completely cancels your flight, you should know well before then.
For Southwest-cancelled flights, call in to Southwest and request a refund. Sometimes, there is push back on a refund from the general reservations number, for complete refunds. If that happens, I suggest contacting Southwest either on Twitter or Facebook, for a refund request.
If your flights are not cancelled by Southwest, you can cancel online. Before doing so, make sure that each ticketed passenger has their own Rapid Rewards account, where the travel funds credits will be deposited into each individual's account. Southwest announced that they will be offering the opportunity to convert travel funds ($$$) into Rapid Rewards points. Unlike travel funds, which only can be used for the passenger named on the ticket, RR points can be used to book travel by anyone. If you explain the situation to the kids' families, perhaps something can be worked out, so that you can use those points for future travel for your family. There has not been a date announced, yet, for the funds to points conversion.
Were you reimbursed by the kids' families for the tickets? That would be a solution, with them having travel funds to use for another trip for the kids.
You could also run this past @SouthwestAir on Twitter, now, for some feedback. Sometimes, if the issue is nicely explained, with any assistance noted as being "greatly appreciated," there could be some accommodation, though don't look for that upfront.
This issue has come up before...you are not alone. It's always better to get a credit card number from non-family when booking travel for them. But, of course, who could have ever imagined this year's happenings?