Is it okay to not have a vapor barrier in lightweight tiny house type construction?
Didn't you ask the same question somewhere else recently? This sounds really familiar.

You should stick to proven wall designs. 2x3s are going to be more difficult to work with. No sheathing will mean virtually no racking resistance against wind. You almost never want a dedicated vapor barrier, especially if you're planning to use rigid foam anywhere (that will result in a vapor trap). Your exterior non-sheathing plan is risky if you don't plan to put housewrap or tar paper under it.

If you really want to do this, here would be my recommendation, inside to out:


Lightweight wood paneling or whatever
2x3s for structural framing with the cavities filled with slightly compressed unfaced fiberglass or mineral wool batts (installed well)
Diagonal steel braces for racking resistance
1+" Foil-faced non-structural foam sheathing (use Polyiso if you live where it's hot; otherwise EPS) over the outside of the studs to act as thermal break, with seams scrupulously taped with high-quality tape to constitute a water-resistive barrier layer
1x3 or 1/2" plywood strips fastened through the foam and into the studs to act as vertical furring strips
siding fastened to the furring strips


This gives you a safe, compact, energy-efficient wall.

Final thought: if you want a weekend camper, that's what you should buy. It will likely cost you far less money than doing all of this yourself.