Vinyl lap siding takes its lumps from quite a few directions in the green building world, despite its versatility, affordability, and durability. But vinyl lap siding is more dependent on weather-resistive and flashing details than almost any other wall cladding.
In general, exterior above-grade walls should have continuous air, water, and thermal barriers.
Installing rigid insulation as the exterior sheathing can accomplish all of these tasks, but you need to account for shear resistance (provide wall bracing) in other ways, and you must provide special details if you expect the foam sheathing to perform as a water-resistive barrier (WRB).
A double-stud wall insulated with dense-packed cellulose is an excellent thermal barrier; however, double-stud walls still need at least one additional air barrier to limit air leakage through the cellulose.
Check out the advanced framing section of Construction Detail Library for alternative techniques to achieve shear resistance, and check out Using Rigid Foam As a Water-Resistive Barrier for more information on walls without housewrap.
For more information on air barriers, see Questions and Answers About Air Barriers