Frank's textual answer seems very tenuous to me. Verb-subject disagreement is far more prevalent than the form here (inconsistent numeration within the same verse when referring to the same subject). So it seems irrelevant to invoke that to justify this construct. Moreover, verb-subject disagreement has literary function (emphasis) which is sorely lacking under that reading. That reading makes the text feel sloppy and redundant.
So while we seem to both agree that the family was burnt along with Achan (the vow in verse 15 clearly states that the perpetrator and all he has will be punished, and so too in verse 24 they clearly take Achan, his family and his possessions), it seems clear to me that the first "him" is a reference to Achan and Achan alone.
I see no necessity in wrapping the singularity in an abstract reference to the entire group as a single entity. Mentioning Achan's punishment first makes emphatic sense (precisely the same reason verb-subject disagreement makes literary sense). He was the perpetrator, so they mention him in his own right.
Moreover, it matches the wording of the vow itself, which mentioned that the person (singular) caught will burn (singular), him and all he owns (a reference to plural entities afterwards). It is more common a construct that two related verses (e.g. a vow and its fulfilment) have similar form, so it makes sense that when the vow is fulfilled it is reminiscent of the original verse.