Passport - premier database from Euromonitor for data and analysis on companies, countries, consumers, and industries worldwide
Also consider searching in business periodical databases - articles may mention the market share -
Search examples - in general, can use the co./product name and add the phrase "market share". But, often, the phrase in the article might be "share of the market". Three database have options to get around this:
in Factiva - coca-cola same market share [phrase is within the same paragraph]
in ABI/Inform Global : coca-cola w/10 market share [phrase is within 10 words of co. name] - if too many results, select "abstract"
in Business Source Ultimate: coca-cola n10 market share [phrase is within 10 words of co. name] - if too few results, select "all text"