Step one is to simply Google up the name of the metro area on Wikipedia. In many markets they have an “Economy” heading that will give you a ton of data.
Step two is the Chamber of Commerce website, which often has a “Top 10 Employers” or “Major Employers” list which includes employee headcount.
Step three is to simply Google up “Top 10 Employers in _____” and see if it pops up somewhere (sometimes in random county data analysis from the state, etc.).
If all else fails, call the Chamber and have someone tell you who they think the top employers are and what makes that metro tick.