Does Congress appropriate 300 billion annually to agencies which cannot legally receive appropriations?
Strictly speaking, no. The monies appropriated by Congress are an act of law and are thus legal, by definition.
However, these appropriations are supposed to be backed by authorizations, and a growing portion of the US federal government's discretionary spending is not backed by authorizations. From the Harvard Law School Federal Budget Policy Seminar paper "Authorizations and Appropriations:
A Distinction Without Difference?" (emphasis mine),
The Constitution grants to Congress the power of appropriation for federal spending. The precise process for exercising that power is derived almost exclusively from internal House and Senate rules. Those rules set out a procedure under which, in general, authorization of federal programs and activities precede, in separate legislation, the corresponding appropriation of actual budget authority. Because these internal rules, by definition, carry almost no statutory or constitutional weight, Congress has little trouble in bending or breaking them whenever it wishes.
The rationale for creating this distinction between authorizations and appropriations was to separate politics (authorizations) from paying the bills (appropriations). The problem is that authorization bills often get bogged down by politics and languish. For example, authorization for the State Department got bogged down by arguments big and small for 15 years. The State Department had no authorization (but most certainly had budget) from 2002 to the end of 2016. The State Department is once again authorized to do its business, at least for a short while, with Senate Bill 1635. This was passed by both branches of Congress and signed into law in December, 2016.
Whether the State Department will be reauthorized to exist next year is dubious. Appropriations will still be granted, even if it is not reauthorized. The State Department is but one part of the US federal government. This repeats many times over throughout the US federal government.