It's more likely that such a request would not get a meaningful response because it would never be routed to anyone with knowledge of such activities. (SAP/SAR/UNACKNOWLEDGED/WAIVED)

I don't know if much would be reported. If you mean a FOIA request to the US Federal government, there are many exemptions, including:

Exemption 1: Information that is classified to protect national security.

...

Exemption 3: Information that is prohibited from disclosure by another federal law.

Exemption 4: Trade secrets or commercial or financial information that is confidential or privileged.

Source: https://www.foia.gov/faq.html

Feel free to try.

FOIA request with whom?

Intel had a ton of firmware leaked a while back and no one found anything particularly juicy.

Many third party researchers have already found backdoors in other parts of Intel's CPUs, such as the Management Engine.

I very much suspect a well funded and motivated actor like the NSA could find mistakes in the microcode without needing to ask for a backdoor directly.