License to use a control suite

I am outsourcing development and design of a website and the vendor plans on using a control suite that requires a license. He claims to have a registered version and has a license to use it. As a customer, should I be worried about the authenticity of the software they use since they are using it on my website?

Yes, absolutely, there's two issues that come into play here (if it turned out that the software was not properly licensed)

1. You could be held responsible by the software creator: While this is unlikely, unless you have an indemnity agreement with the vendor (wherein they accept all liability for the software used on the website), you are ultimately responsible for any liabilities that arise from this
2. It may stop working: Unlicensed software is unsupported and un-guaranteed (obviously) - that means it could stop working at any point and you'd be up * * creek, even more so because a vendor that would knowingly use unlicensed software is probably less reliable and would likely abandon you in that situation.

TL;DR - you should make sure the software is properly licensed: it could save you headaches down the road and its the right thing to do ethically.