Make aioli: Whisk together mayo, Sriracha, lime juice, Old Bay, and 1 teaspoon water. Season with salt, if desired.
Make shrimp filling: In a large skillet over medium-high heat, add oil, onion, bell pepper, and garlic. Cook, stirring occasionally, until lightly golden, about 2 minutes. Stir in shrimp, cumin, and paprika. Season with salt and pepper. Cook, stirring occasionally, until opaque and just cooked through, about 4 minutes.
Assemble burritos: Lay a tortilla on a serving plate. Add 1/4 cup each of the rice and beans. Top with ¼ each of the shrimp filling, cheese, lettuce, pico de gallo, and avocado slices. Sprinkle with chives and drizzle with aioli. Roll burrito tightly and repeat with remaining tortillas and ingredients.
Slice each burrito in half and serve with more aioli, Old Bay, and chives.