After the office of management and budget identifies how federal money should be used, who authorizes the spending?

The Congress

Explanation:

The Congress has power over budgetary and financial matters, through the enumerated power to collect and lay taxes, imposts, duties,  and excises, to provide for the common defense and general welfare of the United States and to pay debts.

The Constitution also allow  Congress exclusively the power to authorize funds.

The house of representative and senate are assigned by the constitution to approve and debate government spending in the spirit of checks and balances as intensified  by the principle of separation of power.