Hypercam is not suited to be used with fullscreen video games, I think. It is for desktop recording. I don't think it is your graphics card as long as you're not using an intel gma.
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But even then maybe you should use MGE. MGE switches Morrowind to DX9 which has many benefits over DX8.
I used fraps and Xfire with success. Xfire is also free.
To use Xfire with MGE, you have to edit the http://wiki.xfire.com/wiki/index.php/Common_data_path
[4312]LongName=The Elder Scrolls III: MorrowindShortName=morwndLauncherDirKey=HKEY_LOCAL_MACHINE\SOFTWARE\Bethesda Softworks\Morrowind\Installed PathLauncherExe=Morrowind.exeLaunch=%UA_LAUNCHER_EXE_PATH% %UA_LAUNCHER_EXTRA_ARGS% %UA_LAUNCHER_NETWORK_ARGS%InGameRenderer=D3D9InGameFlags=USE_PRESENT|ENABLE_MOUSE|USE_DINPUT_MOUSE
InGameRenderer was set to D3D8 normally. We should change that because Morrowind with MGE uses Directx 9.

Mouse doesn't work with xfire gui but keyboard shortcuts are functional and enough. I wonder if some xfire dev or MGE dev look into the issue.
PS. Installing MGE without expansions could be tricky. But I heard it is possible. I mean patching the game to 1.6.