I would drill a couple drain holes from the inside toward the back inside corners. (1/4" hole should work) Then crank up the tounge real high and see if anything drains out.
Typically the roof is made (from top down) of Fiberglass reinforced plastic (FRP) or even aluminum, then foam for insulation, then thin plywood on the inside ceiling for decor. The foam "sandwich" that creates the roof is usually higher density foam, which means, the "foam" doesnt hold water but the surounding material will.
Then check the roof to see where the water is infiltrating.
Hope this helps... Im not a pro, but have cut into a couple of roofs on the pop ups.