What could be causing LAN port unresponsiveness in CHR 6.48.6 on a Supermicro server running VMware 6.7, and how can we resolve this issue without restarting VMware frequently?

What NIC drivers are you using? VMXNET or e1000?

The manual has this to say about e1000 if you are using it:

Warning: We do not recommend using the E1000 network interface if better synthetic interface options are available on a specific Hypervisor!

You also should probably update to the latest 6.xx version which at this time is 6.49.10 at least.

Ask supermicro for a nic fw update, then update the drivers in esxi.
Known issue.