Would utilizing WebSockets cause an issue with obtaining or maintaining PCI compliance?
Not in and of itself; however, like any network connection, it is subject to PCI DSS rules such as the following:
So you'll need to document that any services you set up using WebSocket are necessary and secure; pretty much just the way you would as if it was vanilla TCP.
Again... document that it's needed.
I haven't used WebSocket, but it appears to be just an abstraction layer on top of TCP, and so for PCI purposes does not introduce any inherent insecurities over the set pre-existing in TCP (...until proven otherwise, natch). Obviously for PCI purposes you'll want to pay attention to encryption and make sure that's enabled most places.