Hi @ImperatorMorsus,
We currently don't have a web install version, or the ability to embed a site chat feature into another site. If that's what you're looking for, you can upvote the feature idea here: https://mattermost.uservoice.com/forums/306457-general/suggestions/8810731-implement-a-site-chat-feature
If you're just looking to install Mattermost - I'm not familiar with HostGator, but taking a quick look at it, I think it would depend on what set up their VPS uses.
You can check out our server requirements and install guides to see if any of them match the HostGator set up, and follow the guide that is closest to try installing it.