Heat grill to medium. Once it is hot, clean the grill grate with a wire brush. Just before grilling, oil the grill grate.
Soak 8 skewers in water for at least 15 minutes.
Toss the tofu, oil, lemon juice, and red pepper in a medium bowl. Marinate for at least 15 minutes and up to 8 hours. Add the broccoli, lemon wedges, onion, ½ teaspoon salt, and ¼ teaspoon black pepper to the tofu and toss to combine. Thread onto the skewers and grill, turning occasionally, until the broccoli is tender, 10 to 12 minutes.