Your description isn’t very clear on what exactly you have, or what you need.
It sounds like you have wired NICs in both server and laptop, which will be physically close to each other, but your only connection to the Internet will be WiFi that you don’t control. How accurate is that?
Next question is how do you want them to connect to each other? You can do a P2P wired connection, which is more complicated but fully isolates your traffic. It also means that, unless each device has a separate connection and an appropriate routing config, it won’t be online to the Internet (unless you set up some form of connection sharing). You can also connect them to a router that has no Internet. Simpler than the above, but the same limitations.
You could easily and cheaply get a USB Wi-Fi NIC. The major downside is that all traffic will be going across the wireless connection, both ways. This makes it slow and unreliable.
You can also connect them to a modified router configured as a wireless bridge. DD-WRT and others can be configured in a different way than usual. The wireless router will provide wired LAN ports to your local network, but then use the wireless connection to connect to an upstream WiFi.
None of this has anything to do with Linux, BTW. Once you choose a path, you should be able to implement it in whatever OS (or multiple OSes) you would like. None of it is new or special. You might get more options if you post in the Homelab, Data Hoarder, or Self Hosted communities.