Are the following two lenses fully compatible with the Nikon D7100? Can VR be turned on and off on the one lacking a physical switch? If they are not fully compatible, what exactly works and what doesn't?
AF-S DX NIKKOR 18-55mm f/3.5-5.6G VR II
The lens is fully compatible with Nikon D7100, as virtually every lens released in the last few decades.
AF-P DX NIKKOR 18-55mm f/3.5-5.6G VR
According to the official website:
The number of cameras compatible with both lenses is limited. Even for compatible cameras, firmware update may be required*. Fully compatible models: D5500, D5300*, D3400, D3300*, D500 and later models
Compatible models with limited functions: D5, D810 series, Df, D750, D7200, D7100, D5200, Nikon 1 series with the FT1
Incompatible models: D4 series, D3 series, D2 series, D1 series, D800 series, D700, D610, D600, D300 series, D200, D100, D7000, D5100, D5000, D90, D80, D70 series, D3200, D3100, D3000, D60, D50, D40 series, film cameras
The lens will default to VR on when attached to models released prior to the D3400: D3300, D5500, D5300, D5200, D7200, D7100. These cameras will also require a firmware upgrade to access the menu to turn off the VR.
So yes, you will be able to switch VR on and off, but you will have to upgrade your firmware.
I am unable to find the screenshots of the menu, but basically, there is a menu switch that makes it possible to switch VR. Of course, it's not as simple as doing it directly on the lens. It might be that D7100 can be customized to assign VR to a specific button on the camera.
Make sure to test them
D7100 is great, and using it with a cheap lens may give suboptimal results. I would rather use those lenses on a cheaper body such as Nikon D5200.
If possible, test both lenses before purchasing one of them, not only to be sure that they are compatible, but also to actually see the quality of the photographs you can achieve with those lenses.
In general, remember that lenses should be your primary investment. Bodies are outdated rapidly; however, it is not unusual to use a good quality lens for decades.