I find the VMware HCL incredibly hard to use, but you can start from the IO Devices search, filter for Network, and go from there.

Particularly, most anything from Intel is fine, especially the PRO/1000 series.

It is possible that the LAN port may have died if you are not seeing any activity lights.

You might want to try using a different Ethernet cable first.

If that doesn’t work, check to see if anything shows up in Control Panel > Device Manager > Network Adapters

Also check in “Other Devices” to see if there are any devices with issues.

Finally, you can always buy a replacement discrete PCIEx1 1Gbps Ethernet card for a fairly reasonable price and you can use any of the available PCIE slots in your PC to install to. I don't know where in the world you live, but just as an example

You can get a regular 1Gbps LAN card for under 15 USD

You can get 2.5Gbps LAN card for around 30-45 USD, but it won't make a difference unless the rest of your network is 2.5Gbps capable

You can get an Intel CT Ethernet Adapter which has had a rather large following due to it being around for nearly forever, reputation for being reliable, and given it's age, doesn't require a driver download. Windows picks it up automatically.