Welcome to the community. This is more a question about web programming vs TensorFlow.js itself but I shall try to point you in the right direction. You may want to use some code like this to grab a local file and play it in a video tag (this pen also contains code to load a random TensorFlow.js model and use it to classify a frame of the video when you click the classify button below the video).

Once you have video rendered to canvas you can then pass the canvas to TensorFlow.js pre-made models for classification as usual (or grab a frame from the canvas and convert to Tensor if running on a custom model yourself).
Key things to note:
Your webpage must be served on HTTPS (secure) to access files without issues I believe. You can use Codepen like I did to get free HTTPS page to test on, or:


Glitch.com 1 offers free projects hosted on https. We even have a presence on there ourselves for example code
You could sign up for something like Google Cloud Storage bucket which serves over HTTPS too. But this is a paid service (but will scale to as many users as you need)