Preheat a grill for medium heat. Crush garlic and salt together in a small bowl with a fork. Mix in paprika and cayenne. Stir in olive oil and lemon juice to form a paste.

Combine garlic paste and shrimp in a large bowl and toss until shrimp are evenly coated.

Lightly oil the grill grate. Grill shrimp until opaque, 2 to 3 minutes per side. Transfer to a serving dish, garnish with lemon wedges, and serve.