Template based video storage / online programmatic video editing tool?
As well as @ElderGeek's suggestion of ffmpeg, Adobe After effects can be used to generate bespoke video for the web. It can be fully scripted using Adobe's extendscript on the server and run as a headless process. You can thus programmatically create video templates and alter such things as text, image content, or insert clips, then render and deliver. 

Note that the rendering does take time and CPU (for both ffmpeg and AE). To achieve real-time customisation you may want to consider using the HTML canvas element, which can support video and animation, working on the client side.