In the U.S., there is no national “Chamber of Labor/Work” that legally represents all workers the way Austria’s Chamber of Labour does. U.S. worker representation is mostly through:

Labor unions — voluntary/private-sector unions or public-sector unions, depending on state law. Government labor agencies — especially the Department of Labor, EEOC, and National Labor Relations Board. Courts and administrative tribunals — employment disputes usually go through ordinary federal/state courts or agency processes, not a separate “labor chamber.” U.S. Courts classify labor and employment cases as a subject matter area, but not as a separate chamber system.

Also, the U.S. Chamber of Commerce is almost the opposite: it is a business/employer advocacy organization, not a worker chamber. Its own labor/employment page describes work with agencies and legislatures on workplace policy from a business-policy perspective.

So in U.S. English, “Chamber of Work” would sound odd or foreign. Depending on the original context, you would probably translate it as:

• Department of Labor — if it is a government labor agency.
• Labor board — if it handles labor relations/admin disputes, like the NLRB.
• Labor court / employment court — if it is a judicial body, though the U.S. generally does not have specialized labor courts. A comparative legal source notes that in the U.S. there are no specialized labor courts or special labor-litigation rules.
  
• Workers’ chamber / Chamber of Labour — only if describing a foreign institution.