Do VRAM/Registers get saved when hibernating?

When a computer goes into ACPI S4 state it saves its RAM contents to disk and shuts down.

But what happens to registers, VRAM, or volatile memory inside extension cards or similar?

The TL:DR answer to your question is: Not as a part of the hardware/BIOS S4 implementation, though there is nothing stopping the OS/drivers from doing it.

According to the ACPI specification the Operating System is responsible for initializing all system devices to a usable state after restoring from an S4 hibernation.

ACPI Spec page 7-142 "System \ _S4 State":

• Devices states are compatible with the current Power Resource states. In other words, all devices are in the D3 state when the system state is S4

The D3 state is defined as:

D3 - Off:
Power has been fully removed from the device. The device context is lost when this state is entered, so the OS software will reinitialize the device when powering it back on. Since device context and power are lost, devices in this state do not decode their addresses lines. Devices in this state have the longest restore times.
All classes of devices define this state.

For the driver to function under Windows (or any OS for that matter) then it will have to allocate any buffers necessary and store some quantity of state information. This will be stored in the OS memory space and thus will be stored in the hibernation file.

The OS knows what devices to initialize and likely just calls the registered driver initialization code which will check whether any state information was saved in its internal buffers and restore the card to a usable state.

This just reiterates that the cards themselves know nothing of how to bring themselves out of hibernation state and come up into their default "power on" state until Windows (through the drivers) tells it otherwise. There is no requirement to store the card registers outside of OS dependent drivers which will get their memory space saved by the hibernation process.

When the computer's execution is interrupted for sleeping, all CPU registers for the current executing thread(s) are saved in system RAM, as is the normal procedure with interrupts. So these are saved with the RAM and can be seamlessly restarted on wake.

Video RAM or VRAM is not saved and will have to be re-generated by the OS when it wakes to refresh the display, which it does automatically. So a black screen might result for an extremely short (usually unnoticeable) time.

If a used extension card is withdrawn during the sleep, the process that is using it will fail after wake, as will any CD-reading software if the CD is withdrawn, or any process using an outstanding Internet connection. What happens then depends on how that process was programmed.