What is the statute or authority that exempts Ability One from having an approved accounting system?
There are two separate issues in the question. We'll address CAS exemption first, then adequacy of the accounting system.
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A contract awarded through the AbilityOne program would be exempt from CAS because 48 CFR 9903.201-1(b)(5) exempts contracts in which the price is set by law or regulation; i.e. "Contracts and subcontracts in which the price is set by law or regulation".
FAR 16301-3 requires the contractor's accounting system to be adequate in order to receive a cost-reimbursement contract. This is done to when determining contractor responsibility. But a review of FAR 9.107 (Surveys of Nonprofit Agencies Participating in the AbilityOne Program), para (a) states "The Committee is required to find an AbilityOne participating nonprofit agency capable of furnishing the supplies or services before the nonprofit agency can be designated as a mandatory source under the AbilityOne Program. The Committee may request a contracting office to assist in assessing the capabilities of a nonprofit agency." That means it is up to the Committee to decide if the contracting office will assist with an assessment of the contractor's accounting system.
So in your scenario, there is no requirement for CAS, or an approved accounting system when procuring from AbilityOne.