First, ESXi isn't supported on desktop class hardware, and second that NIC does not have drivers in this build of ESXi.
For a home lab learning purposes, using desktop hardware is fine. But if you plan to run this for business where uptime is important, it is best to go with certified server hardware and to be able to get proper support.
To continue with the Aurora R7, what you need to do is buy/install a network card that is in the VMware Compatibility Guide.
You can select the ESXi version and Network and select particular vendor. Easiest is to look for Intel and Broadcom as there are NIC models that can be bought from online and physical stores. Just be careful that if a price is too good to be true, there is a high chance it is not a genuine and may not work also with ESXi.
Alternative to buying a supported NIC is to use Workstation/Player and create an ESXi VM. You have to enable the "Virtualize Intel VT-x/EPT" so that you can run both 32-bit and 64-bit VMs inside the ESXi VM.
Looks like I'm in this boat too. I bought a Dell Inspiron 13 5379 with quad core processor to run a learning version of ESXi 6.7. Now know that ESXi can't be installed without a 1 GbE network adapter, Dell says the NIC can't be added to this machine.
Could you direct me to more information about setting up the Workstation/Player?
Its what I do. Look at this
it looks pretty good in how to set the up. If you don't have workstation I suggest getting VMUG Advantage, its 200 a year and you get a 1 year license for workstation, vcenter, esxi, and some more.
1. Please make sure your hardware support ESXi 6.7
2. NIC should be supported as per VMware HCL - VMware Compatibility Guide - I/O Device Search
3. Make sure you got the supported NIC Driver- Firmware running , as per VMware HCL - VMware Compatibility Guide - I/O Device Search
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