Can I record game footage from my retro consoles (PS2, Nintendo Wii, GameCube, N64, Sega Genesis) with my AVerMedia capture card?

Yes, but you will need composite/component to HDMI converter. For this purpose we highly recommend our ET111/ET113 converter.
Setup:
Connect your console to ET converter via composite/component cable* and connect HDMI from converter to your AVerMedia game capture card’s HDMI in. Make sure to plug converter to USB power supply.

*If your old console has only composite (AV) output please consider ET111,
if your old console has component output please consider ET113.