tap (verb) to penetrate, open up, reach into, etc., for the purpose of using something or drawing something off; begin to use: to tap one's resources.
tap - Definitions from Dictionary.com
The phrase on green is a unit:
"The (British) Government makes little effort to tap European Community Funds for training women because they discriminate against men, according to the Department of Employment."
Here's how to read it. Because Community Funds are for women only, they, the Funds, discriminate against men, which is why the Government makes little effort to use/tap those Funds.