Where can I look up a companies Herfindahl index? Where can I find Herfindahl indexes on companies (current or past)? Are there any published lists put out by a research institution?
HHI index cannot by definition be on companies. It is the squared sum of market shares of all the companies in a country. So, if you are looking for companies' market shares, you should look at each country individually.

But if you want HHI index on countries (which is a correct way of framing the question) then you can use this WorldBank data: https://tcdata360.worldbank.org/indicators/hh.mkt