I know this exists, we got a tiny little tube of it with our water bed and it works well. It's just clear glue that welds the vinyl but doesn't dry hard.
However I can't find anything that's obviously right when I search. The trouble is that one gets loads of hits on vinyl flooring glue and also loads of hits on glue suitable for hard material (e.g. waste pipes).
I want a simple, one part, glue if at all possible and I need a fair amount of it (maybe 0.5 or 1 litre).