IPWorks provide entire WebSockets library that can work with many programming languages including Pascal. And yes you can get their components to work bot with Delphi as with FPC/Lazarus.
You can find more information about IPWorks WebSockets library here: https://www.nsoftware.com/ipworks/ws/
Granted this is not Freeware software. Delphi licence costs $799 but their components are well known and highly praised in Delphi community.

I personally don't have any experience with them because so far I didn't have any need for sophisticated networking in my projects. But since they do offer a 30 day trial period you probably don't have much to lose if you try them out except your time.