Yes, it's possible with caveats, depending on the specifics of your situation.
WebRTC
With WebRTC, you can establish a real peer-to-peer connection between two clients on a network. They can send data (binary or strings), and media streams (like webcams and microphones). This can even work from a static page.
The catch is that you need some sort of server to help coordinate the connection. Because of the way the WebRTC standard was originally set up, there is some back-and-forth that must occur to set up that peer-to-peer connection. Some method of communicating the signalling between the clients must exist, and this is usually done via web sockets and an intermediary server.
There are some novel alternatives.
In the future, ORTC may solve this issue, allowing a one-shot method for setting up the call, making the server-side requirements easier.
Embedded HTTP Server
You didn't elaborate on the specifics of what device you want to communicate with on your network, so maybe this is a possibility as well. There's nothing stopping your browser from communicating with devices on your LAN. Your static web page can use the Fetch API or AJAX to retrieve data from devices.