While this is correct it is confusing because of how often the term “vinyl” is misused.
Vinyl is a catch-all term that can mean the actual plastic used in something or a variety thin adhesive plastic films. With that in mind:
All PVCs are vinyl but not all vinyls are PVC.
This applies to the actual plastics (there are so many types of vinyl compounds) as well as adhesive or heat transfer films that are marketed as “vinyl”. Even Glowforge calls their proofgrade “vinyl” when we know it’s made of polyurethane.
With any plastic it’s crucial that you understand what it’s made of.
As for this bit:
I suspect they mean “it might catch on fire” or “it is unsafe to breathe the fumes”. This is true of literally everything you laser, so that’s not a specific concern. This big risk, as always, is PVC. Their site seems to indicate that it’s made of polyurethane which lasers well and does not emit compounds that will damage your machine.
I’d push back on Carolyn and find out why specifically she said that.
For more context on what materials are dangerous to put in your laser and why, check out number four on this list:
If anyone’s still reading and wants to see how many vinyl compounds there are (it’s a huge family of organic chemicals), check out this wiki page:
Behold, the vinyl group: you got one or more of these and you’re a vinyl 
