Preheat oven to 400 degrees F (200 degrees C). Prepare a baking sheet with cooking spray.
Heat a large skillet over medium-high heat. Cook and stir turkey and onion in the hot skillet until turkey is completely browned, 5 to 7 minutes.
Stir sloppy joe sauce into the turkey and onion mixture; simmer until slightly thickened, about 5 minutes. Remove skillet from heat and stir Cheddar cheese into the turkey mixture until melted.
Lay an egg roll wrapper onto a flat surface. Spoon about 2 tablespoons of the turkey mixture down the center of the wrapper making a line that reaches to within an inch of 2 opposing corners. Fold the near corners over the filling so the points overlap the filling. Fold one of the remaining corners over the filling and roll to form a log. Repeat filling and rolling with remaining filling and wrappers until all filling is used. Brush olive oil over the rolls to coat and arrange onto the prepared baking sheet.
Bake rolls in the preheated oven until lightly browned and crispy, 12 to 15 minutes.