The Fed Governors are unelected and have a tremendous amount of influence over the economy. Is this system compatible with a democratic form of government

No, it doesn't.

Explanation:

Given that democracy is a form. of government or system of governance in which those in public offices or holding political positions are fully elected by the people or citizens of that particular country or state.

Also, democracy involves certain principles among which is operations of powers between all levels and branches of government and then check and balances between branches of government.

Hence, where Fed Governors are unelected and have a tremendous amount of influence over the economy. It is believed to be UNDEMOCRATIC FORM OF GOVERNMENT