"Running an update where it pulls new code" is definitely an example of phoning home, and it opens the door to all sorts of vulnerabilities, such as the one featured in the Apple vs the FBI case.
So normally only the build servers contact Intel. Source-based distributions may. But even then the source files are verified with cryptographic hashes (which are in turn signed by the maintainers' private keys).
Edit: I may have misread your question. I thought you were asking by implication if the microcode blob is pulled after the install. If not, then: No. Updates happen when the package manager decides.
https://www.kernel.org/doc/Documentation/x86/early-microcode...
https://gitlab.com/iucode-tool/iucode-tool
https://downloadcenter.intel.com/download/26400/Linux-Proces...