This is how to get market capitalization with Alpha Vantage API:
Market cap is part of the company overview dataset, which you can access using function=OVERVIEW.
Other two parameters for the overview API URL are symbol (the stock symbol, e.g. "AAPL") and apikey (your Alpha Vantage API key).
The complete URL is:
https://www.alphavantage.co/query?function=OVERVIEW&symbol=IBM&apikey=demo
The API URL returns JSON format.
Market cap is under the key MarketCapitalization.
The company overview also includes total number of shares (SharesOutstanding key), which you can multiply by current share price to get market cap alternatively.