As you can see from the other answers there are a lot of Delphi libraries to simplify network programming. However, AFAIK they all provide a friendlier layer over the WinSock API (on Windows, over the standard socket API on other OSs) and thus are not able to do anything that the WinSock API isn't able to do. In particular they may or may not support raw socket access, but if the OS fails to send data over the socket the libraries will only return errors as well.
For more information about the limitations of raw sockets on most recent Windows OS versions see this MSDN page, in particular the section "Limitations on Raw Sockets". It states quite clearly that
To get around these issues, it may be required to write a Windows network protocol driver (device driver) for the specific network protocol.
Neither of the Delphi network programming libraries comes with its own network protocol driver AFAIK. And neither will help you with permission issues on limited user accounts, or the interference of the Windows Firewall.
As you mention WinPcap in your question you know it is capable of doing what you want. There are Delphi wrappers for it, like Magenta Systems Internet Packet Monitoring Components. However, as a C++ programmer you may be better off to use the library directly from C++, as Delphi adds nothing in regard to the socket programming you want to do. In particular you won't need the extra layer to hide platform differences that some of the Delphi libraries provide, as you will be programming directly against the libpcap API.