Hi, Swordy!
Is the GAO tasked with determining violations of law under their charter?
Are they part of the DOJ?
No, they are not. They aren’t part of the Executive Branch. From the GAO.gov website:
The U.S. Government Accountability Office (GAO) is an independent, nonpartisan agency that works for Congress. Often called the "congressional watchdog," GAO examines how taxpayer dollars are spent and provides Congress and federal agencies with objective, reliable information to help the government save money and work more efficiently.For example, we identified $214.7 billion in financial benefits in fiscal year 2019—a return of about $338 for every $1 invested in us. We also identified 1,418 other benefits that led to program and operational improvements across the government. Read the press release about these record numbers.
So, no, they do not have the power to determine if anyone has broken the law. That is not their charge. This is a gratuitous assertion.