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:
1.1.5.a Verify that firewall and router configuration standards include a documented list of services, protocols and ports necessary for business—for example, hypertext transfer protocol (HTTP) and Secure Sockets Layer (SSL), Secure Shell (SSH), and Virtual Private Network (VPN) protocols. 1.1.5.b Identify insecure services, protocols, and ports allowed; and verify they are necessary and that security features are documented and implemented by examining firewall and router configuration standards and settings for each service.
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.
1.2.1.a Verify that inbound and outbound traffic is limited to that which is necessary for the cardholder data environment, and that the restrictions are documented.
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.