Heat oil in a large pot over medium heat; stir in onions. Cook and stir until the onions has softened and turned translucent, about 5 minutes.
Stir in curry powder, cumin, turmeric, and coriander, adding water as needed so nothing sticks to the bottom. Add vegetable broth, sweet potatoes, white potatoes, and chickpeas. Bring to a boil, then reduce heat and cook until potatoes are soft enough to blend, about 20 minutes.
Blend soup with an immersion blender until desired consistency. Season with salt and pepper.