I am currently working on a Chromebook (it's the only thing available to me at the time) which is managed and does not have the ability to install Chrome Extensions (such as https://chrome.google.com/webstore/detail/secure-shell/iodihamcpbpeioajjeobimgagajmlibd?hl=en) or enable Linux Beta. Is there a website that will allow me to make SSH connections purely client side so that I can connect to a LAN server, or a Vanilla JS library that I could use to write my own?
The APIs provided by browsers allow network connections to be made through a very limited set of protocols (such as HTTP and WebSockets). They don't support SSH or the raw sockets needed to write an SSH library from scratch.
The only way to do something like this would be to run a server which converted SSH into something else (e.g. with the server establishing an SSH connection and then relaying the data to and from it over a WebSocket).