Well, no one actually saw then "take off" that we know of. The kids weren't at the apartment when the police came by to check on them (or at least they weren't there that the police know of). That's when they said they were in AZ. Beyond that do we really know they were or weren't there.
Something else struck me. The address where they did the welfare check and presumably where the kids were believed to live with their mother and presumably Chad was an apartment in Rexburg. But Chad and his deceased wife based on something else I saw lived north of Rexburg, kind of in the Salem area, but just across the Madison/Fremont county line. (Hence the reason the Fremont police were involved in some of this.)
Given the timing of everything and the potential that the apartment might have a lease that wasn't up yet I can see how they might have continued maintaining both houses. I guess Chad might have moved in with Lori and if he was renting the house it would be out of the picture.
But what are the odds that they did have access to both places and were spending most of their time at the house out in the sticks? Maybe when the police came by looking for them at the apartment the kids were actually at the other house. Throw in some paranoia about the government or whatever and I can Chad and Lori claiming they were in AZ since that wouldn't be easily and quickly proven or disproven. They could have then grabbed the kids and taken off. This is obviously MOO or whatever acronym means throwing out unfounded theories that could maybe be.