Cut the tofu into 4 thick slabs crosswise. Blot well between paper towels. Cut each slab into 6 large dice.
Alternate the chunks of tofu, bell peppers, tomatoes, and mushrooms on each of 8 bamboo skewers, leaving about 1 ½ inches of skewer at the bottom, and filling the entire skewer as close as possible to the top.
Wrap the ends of the skewers with aluminum foil.
Combine the oil, soy sauce, wine, and basil in a small bowl and stir together.
Heat half of the mixture on a griddle (either round or square) on the stovetop.
Arrange 4 of the skewers on the griddle with the longer wrapped skewer ends hanging off the edge. Cover skewers and heat over medium-high heat, turning frequently, for 8–10 minutes total or until the tofu and vegetables are touched with a few golden brown spots.
Move the skewers to a plate and cover with foil to keep warm. Repeat with the remaining skewers.