try disabling the firewall on Vmix system.
Are both systems on shared network?
Also possible your system is putting network card to sleep.
Check advanced power settings.

You will need to get your network admin to allow traffic in both directions. If you are the network admin, look at the port fowarding, routing tables etc on your router.

Your problem is the same as with all network services running on a local computer: in order to reach the web-interface several gates need to be opened through firewalls, routers, etc.

From the inside out:

You are running vMix on a windows pc that is protect by at least windows' own firewall. In this firewalls settings you have to allow incoming traffic to vMix' web-port, usually that is 8888 (or whatever you choose in settings->web).

Most windows pc also have some kind of "personal firewall", usually bundled with antivirus software. There you also must allow incoming traffic to vMix' web-port; sometimes you also need to allow the vMix executable to access the internet at all.

These two steps are enough to access vMix from other computers in the same network.

If you want to access the web-interface from another network, there is another barrier to cross: usually there is some kind of nat-router, separating different networks (like wlan, powerline, lan in the same house) and translating network adresses. You have to open a path through that router, usually called "port forwarding", so that requests sent to the routers ip at a specific port are forwarded to the pc running vMix. Be aware that in this scenario you have to connect to the routers ip (which is visible from the outside network) instead of the vMix pc's ip (which is invisible to the world behind the router!).

The same principle applies if you want to connect through the internet, but some isps don't allow incoming traffic on some ports (or even all ports...).

There is a large number of tutorials how to run a web-server on your local machine and make it accessible from the outside - all tipps therein apply to vMix' web-interface as well.