Preheat oven to 425 degrees. On a large rimmed baking sheet, toss mushrooms with 1 teaspoon oregano, 1 tablespoon oil, and 1/4 cup water; season with salt and pepper. On another rimmed baking sheet, toss zucchini and onion with remaining teaspoon oregano and tablespoon oil; season with salt and pepper.
Place both sheets in oven. Roast, tossing occasionally, until vegetables are browned and fork-tender, 25 to 30 minutes (zucchini may cook faster than mushrooms).
Meanwhile, in a small skillet over medium-high heat, warm tortillas according to package instructions (they should be lightly browned but still soft). Wrap loosely in a clean kitchen towel to keep warm.
To serve, fill each tortilla with mushrooms, vegetable mixture, shredded cheese, and salsa.
To clean portobellos, snap off stem, then use a small spoon to gently scrape (then discard) the dark gills from the inside of the cap.