18F was a business unit of the General Services Administration, which was created via 41 U.S.C. 251, the Federal Property and Administrative Services Act of 1949. The act transferred the function of the Federal Works Agency and the Treasury Bureau of Federal Supply to the GSA. The agency is directed and authorized to perform a variety of tasks, from real estate, to acquisition, to information technology and telecommunications.
31 USC 1535, The Economy Act, authorizes Federal agencies to purchase from each other in the interests of economy. Obvious application is that rather than have the Social Security Administration lease an office or build a building, they sublease or lease a facility owned by the GSA so Federal demand can be aggregated. 1 big lease is cheaper than 12 little ones.
In the case of 18F, having a consultive entity within the GSA maximizes the value of procurements made by GSA and other agencies. When congress appropriates money to say, the Department of Labor to perform a function, DOL may choose to engage 18F to deliver or assist, avoiding additional procurement and taking advantage of investments and capability already built.
Congress in 1949 recognized that basic concepts like shared services, aggregation and mission focus deliver value and promote efficient operations. The current regime's corrupt interest is obvious to anyone with a brain.
18F is not the only department that is being shut down.
The current iteration of USAID was created by an act of Congress. Unless the President's office has been given the power to create and pass legislature at some point in the past five weeks, only Congress can unmake it.
That's what makes it illegal.