According to the White House's argument, the President can order it himself.
Back in 1965, LBJ signed the Higher Education Act, allowing student loans. In 2003 Dubya signed the HEROES Act, which modified the earlier act to allow the Secretary of Education to waive or modify any student loan agreement for anyone who has "suffered direct economic hardship as a direct result of a ... national emergency." The President is the Secretary of Education's boss, and can direct him to do so.
Naturally, many are opposing it, presumably because they don't think this is a "national emergency" in the manner the lawmakers in 2003 intended, so it will be up to the courts to decide. Just today, a federal court put it on hold pending arguments, which was to be expected. Honestly, it could go either way, but in the meantime, 22 million people have signed up so far for $10K-$20K worth of forgiveness. Each.