Which government agency primarily deals with domestic policy issues?

You can call them domestic policy council or the Congress.

Explanation:

Domestic policy covers a wide range of areas, including business, education, energy, healthcare, law enforcement, money and taxes, natural resources, social welfare, and personal rights and freedoms.

"Domestic Policy Council," as it is known today, was created by Executive Order in 1993. Under President Bush, the Domestic Policy Council oversees major domestic policy areas such as education, health, housing, welfare, justice, federalism, transportation, environment, labor and veteran's affairs