Line a baking sheet with parchment paper and set aside.
Combine half of the shrimp, 2 tablespoons of mayonnaise, 1/4 teaspoon of salt and a pinch of black pepper in the bowl of a food processor and process until mostly smooth. Add the remaining shrimp and pulse until the shrimp is mostly chopped into large pieces. Carefully transfer the shrimp mixture onto the prepared baking sheet. Wet your hands and lightly form the shrimp mixture into a disc (wetting your hands will prevent the shrimp mixture from sticking to them). Divide the disc into 4 equal amounts. Use your hands to form the mixture into 4 sausages or "shrimp dogs" approximately 6 inches long and 1 inch wide, wetting your hands as needed. Cover with plastic wrap and refrigerate while you make the grilled vegetable relish.
Prepare a grill or large grill pan for cooking over medium-high heat. Thread the tomatoes onto the wooden skewer, leaving only 1/2 inch of the skewer revealed on each end. (This prevents the skewer from having to be soaked in water before grilling.) Put the skewered tomatoes onto a baking sheet along with the onions and pickle slices. Brush all of the vegetables liberally with 1 tablespoon oil and sprinkle liberally with salt and pepper.
Lightly oil the grill grates. Arrange the skewered tomatoes, onions, and pickle slices on the hot grill. Cook, using a pair of tongs to rotate the tomatoes occasionally so that they cook evenly, until the tomatoes are lightly charred in spots and have started to burst, 6 to 8 minutes. Cook the onions until lightly charred and the flesh is tender, turning once, 6 to 8 minutes total. Cook the pickle slices until lightly charred and distinct grill marks form, turning once, 6 to 8 minutes total. Transfer the grilled vegetables to the same baking sheet that they were on before. (Leave the grill on.)
Cool the vegetables slightly, then quarter the tomatoes and chop the onions and pickle slices into small pieces. Place the cut vegetables into a medium mixing bowl. Add 1 tablespoon of oil and the pickle brine. Stir to combine and season to taste with salt and pepper. Reserve.
Remove the shrimp dogs from the refrigerator and brush them liberally with the remaining tablespoon extra-virgin olive oil. Sprinkle lightly with salt and pepper. Use your hands to carefully arrange the shrimp dogs on the grill and cook, using a pair of tongs to rotate the shrimp dogs occasionally, until lightly charred on all sides and just cooked through, 6 to 8 minutes total. (You will know that they are cooked through when the texture turns from soft to springy and a cake tester inserted through the center comes out hot to the touch.) Transfer to a plate or baking sheet.
Gently open up each of the 4 hot dog buns, being careful not to unhinge them. Arrange the hot dog buns cut-side down on the grill and cook until lightly charred and warmed through, about 1 minute. Smear the inside of the grilled buns with the remaining 2 tablespoons mayonnaise.
Place a shrimp dog on each hot dog bun, then spread a spoonful of the reserved relish on top of each. Serve with any remaining relish on the side.