I don’t know anything about this device, but I do know the VPE does have the code for it and you will need to take control of the VPE and add the code to drive it. That is if eshome supports its.
I’m no expert, but I’m 96.25% certain that all that adapter does is allow you to extend the grove port over existing phone or ethernet cable (for scenarios where you want a grove sensor in a different room).
It’s not going to magically transform your VPE into an ethernet capable device by using the 2 I2C data lines. Also, that VCC pin injecting 5v back into your router might fry it, so don’t risk it.
I believe the part you need is one of these. Then wire the pins to the correct ones on the Voice PE. I haven’t tried this so don’t have any experience with it.