Azure Maps is a global geospatial service and FIPS codes are USA Census data specific. As such, FIPS codes are not used or supported by Azure Maps. Note that county level FIPS code are 5 digital numbers which is commonly used as zip codes/postal codes in several countries, and since geocoders are address focused, they will most likely interpret FIPS codes as such.
That said, there is a fairly small list of FIPS codes in the US and it wouldn't take much to create a simple look up table in your database and then joining it with your business data. This would likely work really well.