Why does my utility sink faucet vibrate and pound loudly?
Although the faucet looks to be an older style faucet it looks to me more like a model that is currently still in production and still available. Based not only on the photo but also on the video.

That faucet has a cartridge instead of a stem valve like older faucets might have. Could be as simple as something stuck in the cartridge or the cartridge has worn out.

The first thing I would do is try and swap the hot and cold water cartridges to see if the noise follows the cartridge. If it does, replace the cartridge.

If that's not the problem make sure everything is tightened down properly where the spout meets the faucet body and check that all the fasteners for the piping are secure.

There are a couple of manufacturers that make that style of faucet which you'll need to contact for a replacement cartridge. Home Depot currently has one from Glacier Bay which would be my first guess since they've been around a while. Here's a link to the faucet on HomeDepot.com for reference.