In a food processor, pulse oyster crackers into crumbs, then transfer to a large bowl. Add the cornichons, capers, scallions and tarragon to the food processor and pulse into small pieces. Add the mayonnaise, lemon juice, mustard and pepper and pulse to combine.
Remove 1/2 cup tartar sauce to a bowl, cover, and refrigerate. Pour the remaining tartar sauce into the bowl with the cracker crumbs and gently fold together with the egg and crabmeat. Use a 1/4-cup measure to form the mixture 16 crabcakes.
Heat a large nonstick skillet over medium-high heat. Add the olive oil and, working in batches, cook the crabcakes until golden brown, about 1 minute per side. Serve with a dollop of the reserved tartar sauce and lemon wedges.