Make the kebabs: Combine the garlic, shallot, ginger, lemongrass and cilantro in a food processor and pulse until finely chopped. Add the lime zest and juice, fish sauce, soy sauce, sugar and vegetable oil and process until a coarse paste forms. Put the chicken in a large bowl and add the ginger paste; toss to coat. Cover and refrigerate 4 to 6 hours.
Meanwhile, make the pickled vegetables: Bring the vinegar, 1/4 cup water, the sugar and 1 teaspoon salt to a simmer in a small saucepan over medium heat, stirring, until the sugar and salt dissolve. Let cool slightly. Combine the carrots and cucumber in a small bowl and pour the brine over the top; press the vegetables to submerge. Refrigerate until chilled, 1 to 4 hours. Drain, then stir in the cilantro.
Make the spicy mayonnaise: Stir the mayonnaise, lime zest and juice and Sriracha in a small bowl; season with salt. Refrigerate until ready to serve.
Preheat a grill to medium. Drain the chicken, brushing off any large pieces of the ginger paste. Thread the chicken, daikon and jalapeños onto ten 12-inch skewers. Season with salt.
Brush the grill grates with vegetable oil. Grill the kebabs, turning often, until the chicken is cooked through and well-marked, 12 to 15 minutes. Remove to a platter and sprinkle with cilantro and Thai basil. Serve in baguettes with the pickled vegetables, spicy mayonnaise and lime wedges.
If you use wooden skewers, soak them in water for at least 30 minutes so they won't burn.