Is it possible to download zip code or ZCTA-level data?

Please clarify.

Microdata are observation-level data: housing unit, household, family, person.

ZIP/ZCTA is aggregated data. You won’t have ZIP in microdata (not in the public version at least; it may be there in the restricted data).

You can aggregate the tract or block-group level tables from American FactFinder using the Missouri Census Data Center geography conversion tool Geocorr 2018 - MCDC.