What was the first worms game? Does it even exist?

Of course it exists. It's called 'Worms'.

Is there any sort of website for it?

Is there any sort of website for it?
I think it was invented before the internet became mainstream.

At this site (http://worms.team17.com/index.html?page=prev) there was a dead link to a Worms sub-page for quite sometime. But I don't know if it ever existed.
IIRC, I sent a PM to Sel (or was it still APJ?) and asked about it.
Later the link was removed.

These pages are excellent sources of info for the old games:
http://www.strategyplanet.com/worms
http://www.dream17.co.uk

Technically, the first worms game was Total Wormage, which Andy submitted to an Amiga compo and lost, but which Team17 took and released as Worms.

There used to be an official Worms site, back in the old OLD Team17 website. I'd give you a link but archive.org seems to be down.