Crescent and Bainbridge offer adhesive boards such as foamcore, in 3/16" thickness. They also have thinner, but heavier clayboards with adhesive already applied on one side. Many framing wholesale stores (assuming you have a State Sales Tax number), will sell you individual sheets (32x40).
I've done projects like this before, and can recommend gang printing & mounting rather than doing so individually.
I'd suggest gang-printing your 5x7's on whatever the largest size paper your printer can handle (13x19, 17x22, etc), then mounting that sheet onto a same size adhesive board, then cutting down to 5x7 size. Note - the 3/16" foamboard with adhesive cuts very easily with one stroke, and is super lightweight.