Whip the heavy cream, sugar and vanilla in a large bowl with an electric mixer until firm peaks form.
Remove the ice cream sandwiches from their wrappers. Place 6 ice cream sandwiches end to end in a 2-by-3 rectangle on a large flat serving plate. Cover with a thin layer of whipped cream. Stack 6 more ice cream sandwiches in the same manner on top; frost again. Repeat the process until all 24 sandwiches are stacked in one large rectangle. Cover the entire cake with a layer of whipped cream.
Place half of the cookies in a large resealable plastic bag and crush them using a rolling pin. Sprinkle the crumbs around the bottom and top edges of the cake and place some on top of the cake in the center. Arrange 5 whole cookies on top of the cake. Serve immediately or freeze until party time (up to 2 days ahead).