Prepare the vegan tzatziki by combining vegan yogurt, grated cucumber, minced garlic, half the lemon juice, and dill in a small bowl. Refrigerate until ready to serve.
In a large mixing bowl, whisk together 2 tablespoons of olive oil, remaining lemon juice, oregano, smoked paprika, garlic powder, salt, and pepper.
Add the seitan strips to the marinade and toss until evenly coated. Let marinate for at least 15 minutes.
Heat 1 tablespoon of olive oil in a large skillet over medium-high heat.
Add the seitan to the skillet and cook for 8 to 10 minutes, stirring occasionally, until the edges are browned and slightly crispy.
Lightly toast the pita breads in a dry pan or oven until warm and flexible.
Assemble the gyros by spreading a generous dollop of tzatziki on each pita.
Distribute the seitan strips evenly among the pitas and top with sliced red onion and tomatoes.
Fold or roll the pita tightly and serve immediately.