Simplicial commutative monoids can fail to be Kan complexes. I don't know if this also fails assuming they are cancellative (the usual proof for simplicial groups does not seem to work). We say that a commutative monoid is cancellative (a.k.a. integral) if the map into its group completion is injective. So the question is:
Is every simplicial object in the category of cancellative commutative monoids a Kan complex?