Will ethernet work off of Thunderbolt/mDisplayPort/Ethernet adapter?
You seem to have your terminology mixed up, which makes it difficult to answer your question.

Mini-DisplayPort is a standard for transferring display signals only. There exists no such thing as a Mini-DisplayPort to Ethernet adapter. In fact what you have is probably a Thunderbolt to Ethernet adapter. Thunderbolt uses the same physical plug as Mini-DisplayPort, but it is completely different thing.

When you say your laptop requires Thunderbolt, I think you're mistaken again. The picture you have included shows what appears to be a Thunderbolt 3 plug. Therefore I think your laptop really has Thunderbolt 3 ports, and not just Thunderbolt.

Again when looking at the photo, it seems to me that you are mistaken in saying that you have a "Mini-DisplayPort to Ethernet" adapter at all - i.e. it's not even Thunderbolt. The way you have taken the picture makes it impossible for us to see which plug goes with which adapter unfortunately. I do think I recognize the product as being a USB-C to Mini-DisplayPort adapter. If that is the case, you cannot use this adapter at all for this purpose.

You can buy a Thunderbolt 3 to Thunderbolt 2 adapter for connecting the ethernet adapter to your laptop:

https://www.apple.com/shop/product/MMEL2AM/A/thunderbolt-3-usb-c-to-thunderbolt-2-adapter

However it's probably cheaper to just buy a USB-C to Ethernet adapter in the first place. It could be something like this:

https://www.amazon.com/Ethernet-Thunderbolt-Gigabit-Network-Compatible/dp/B07XTGKP5M/