I personally don't think it would be that hard to find someone to care for the child, especially among Latter-day Saint communities who are generally very helpful and giving, at least for a little while. I've been asked by friends to care for their autistic special needs child who requires medications, special attention, and constant supervision while they went on vacation for an extended period of time. I did it because they are friends and had asked me to, and because it was the right thing to do. Granted their child wasn't in the news, listed as a missing and endangered person, with several law enforcement agencies, including the FBI, actively searching for them.
But even still, if Lori and Chad have surrounded themselves with people and friends who are anti-government, conspiracy theorist, end of the world doomsday preppers, who share the same ideologies as them, it might not be out of the realm of possibility for them to find someone they can sweet talk into hiding and caring for the kids. I mean look how many people we've seen come out of the woodworks to defend and support Lori and Chad, even rabidly and mockingly attacking those searching for the kids and answers. This group ain't entirely normal. Moo.
Edited to be more legible.