Finely chop mushrooms, onion, celery and garlic in a food processor. Heat oil in a large skillet over medium-high heat. Add the vegetable mixture and cook, stirring occasionally, until the liquid has evaporated, 6 to 8 minutes. Transfer to a large bowl and let cool for 10 minutes.
Preheat oven to 450 degrees F. Line a large rimmed baking sheet with foil and coat with cooking spray.
Add breadcrumbs, carrot, cheese, hot sauce, vinegar, salt and pepper to the cooled vegetables; stir until combined. Add chicken and mix gently to combine (do not overmix). Form into 30 meatballs (a scant 2 tablespoons each) and place on the prepared baking sheet.
Bake the meatballs until an instant-read thermometer inserted in the center registers 165 degrees F, about 15 minutes.
