International trade and production data sources:

UN Comtrade: https://comtrade.un.org/  
BACI bilateral trade data (CEPII): http://www.cepii.fr/CEPII/en/bdd_modele/presentation.asp?id=37  
FAOStat: http://www.fao.org/faostat/     
International Trade and Production Database (USITC):  https://www.usitc.gov/data/gravity/itpde.htm

Domestic trade data sources:

Commodity Flow Survey (US Census): https://www.census.gov/programs-surveys/cfs.html
Freight Analysis Framework: https://www.bts.gov/faf

Free input-output (supply-chain) databases:

US tables (BEA): http://www.bea.gov/industry/io_annual.htm
WIOD tables: http://www.wiod.org
WIOD environmental accounts: http://www.wiod.org/database/eas13
OECD IO database: https://www.oecd.org/sti/ind/input-outputtables.htm
EXIOBASE: https://www.exiobase.eu
EORA: https://worldmrio.com