Cook rice noodles according to package instructions. Drain and set aside.
In a large skillet or wok, heat vegetable oil over medium heat. Add onion, garlic, bell pepper, and mushrooms. Cook for 5 minutes, until vegetables are softened.
In a small bowl, whisk together coconut milk, Thai red curry paste, soy sauce, and brown sugar. Pour the mixture into the skillet with the vegetables.
Bring the mixture to a simmer and cook for 5 minutes, stirring occasionally. Add fresh spinach and cooked rice noodles to the skillet. Toss to combine and cook for an additional 2 minutes, until the spinach is wilted.
Serve the vegan Thai curry noodles hot, garnished with cilantro leaves and lime wedges.