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)
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.