Can VLC output video through HDMI adapter to TV on first gen iPad?
According to Apple's documentation, the AV Adapter for iOS supports mirroring only on the iPad 2. On the iPad and iPhone, video out has to be supported by the app itself, whether Apple designed it (YouTube) or a third party app (Netflix, Hulu).

Because VLC for iOS hasn't been updated in quite a while (indeed its no longer available on the App Store for legal reasons), I wouldn't be surprised that the VLC app just does not support video out, although I don't know for sure if that's the case.

There is recourse if you jailbreak, DisplayOut, which enables mirroring on VGA, HDMI, component, or composite outputs for the original iPad, iPhone, and fourth gen iPod Touch. If you are able to install this you would then be able to output VLC on your iPad via the HDMI adapter.