On the thickest part of each side of the snappers, cut a 1/2-inch-thick incision along its length. Season the snappers inside and out with salt and pepper and tuck 3 tarragon sprigs inside each fish's body cavity.
Cut 6 pieces of aluminum foil, each large enough to completely enclose each fish. Preheat the oven to 450 degrees F.
Arrange half of the julienned vegetables in beds in the center of each piece of foil and place a fish on top. Drizzle the fish generously with olive oil. Arrange the remaining vegetables on top, then cover with lemon slices and the remaining herbs.
Close the foil around each fish, double pleating the edges together to form airtight packages. Place the packages on a baking sheet and bake them until they have puffed up like balloons, about 15 minutes.
Serve each foil package on a dinner plate. Let each guest carefully open the package to eat the fish, easing the fillets off the bone along the top incision and then pulling out the spine and ribs in a single piece.