Fingerprints are required in a number of states to get a CPA license. Under California Board of Accountancy Regulation 37.5
A licensee applying for renewal as a certified public accountant or
public accountant who has not previously submitted fingerprints as a
condition of licensure or for whom an electronic record of the
licensee's fingerprints does not exist in the Department of Justice's
criminal offender record identification database shall successfully
complete a state and federal level criminal offender record
information search conducted through the Department of Justice by the
licensee's renewal date that occurs after December 31, 2013.
This is the implementation of the statutory requirement that the licensee be of "good moral character". You find this in Texas (§901.169). In Georgia, this is a consequence of Board Policy 19:
The Board designates the National Association of State Boards of
Accountancy as its agent for the processing of applications for an
initial license as a CPA, a reciprocal license as a CPA, and for
reviewing applications for the initial registration of a firm for the
practice of public accountancy.
Georgia also has a "good moral character" requirement, Rule 20-3-.08, but in this case there is no state-level requirement for fingerprints. Instead, this results from the policy making NASBA the agent for the state, and NASBA's determination that they need fingerprints to ascertain the identity of the individual taking the exam.
To get to first base with a legal challenge, you need a reasonable legal basis, not just a dislike of having a fingerprint taken. Given that pictures and signatures are required for many things (passports, driver's license, university ID) providing some concrete evidence of who you are is not unreasonable, and since the procedure is non-invasive (not a blood test) this is not a search at all. NASBA provides ample reasons in support of using fingerprints, so without a strong argument that your constitutional rights are being over-burdened, you have no basis for a legal action. And there no existing opt-out procedure.