No, you don't need a vapor barrier. That would be a bad idea because it would trap moisture which allows mold to grow.

Will it be a conditioned space? That is, will you add heat and stay in there a while? I am asking because living and cooking create moisture that must go out somehow. Vapor barriers for houses slow moisture, not block it.

Rigid foam creates an essentially waterproof surface. OK, you might have some gaps, but not much vapor will move through. That is good and bad. The foam must be adhered closely to the inside plywood or moisture can collect in the space. If you have moisture for any time you have mold and wood rot.

Let's assume you install it well with foamboard adhesive. If you have a tight building, you will need to create circulation from the outside. Barns are not known for being tight.

Even so, for your application I don't see any advantage of using foamboard instead of paper-faced fiberglass batts. Fiber insulation is less expensive per R-value than rigid foam, and the kraft faced batts are easier to install. FB must be cut to fit between the studs, which never have exact spacing. Batts are more forgiving- they squish or fluff a bit.

I have insulated with rigid foam and fiberglass. The only reason I used the FB was below grade, glued to the concrete walls. The previous owner had used fiberglass batts then covered the interior with plastic sheet, which completely stopped vapor movement. Bad idea. It was waterlogged for years. I ripped out two tons of wet, moldy fiberglass.