I want to edit people's videos but need a way for customers to upload them.
https://www.filepicker.io/ is the best thing that you can integrate with this app which suits your requirement. If you are the owner of the app, you can integrate it very quickly. If not, you can suggest the app owner to do it
Feel free to call me if you need more clarification. I am consulting architect for few companies in US and India on mobile, web and windows/linux/mac apps.