Its funny to see a prospective attorney ask the question in a way that you have. While you might be able to pursue an ADA action, it wouldn't be a breach of contract action in federal court. Anyway, it sounds like the State Bar did provide you with an accommodation. Still, you could consult with an attorney experienced in ADA and matters such as this and see if he or she thinks you have any recourse, although I don't think you do...
THESE COMMENTS MUST NOT BE CONSIDERED LEGAL ADVICE. Comments made on websites such as Avvo.com are provided for information purposes only, and you should not base a decision to act or refrain from acting based upon this answer. The only way to determine how the law may apply to your particular situation is to consult with an attorney licensed in your jurisdiction. Answering this question does not create an attorney-client relationship or otherwise require further consultation.
My first response to your question is why would you want to go to all the trouble, expense and angst to sue the Bar rather than simply take the test in six months and pass it? However, I see this happened two years ago, so I must conclude that you either stopped trying, passed it, or took it several times and failed. If you stopped trying you failed to mitigate. If you passed it, your damages are limited. If you failed it several times, it suggests you have no damages.
Furthermore, you state no basis for alleging a contract. The ADA creates a statutory basis for a claim, not a contractual one. Unless you had a contract that you entered into separate and apart from the Bar's statutory duties under the ADA, I see no viable contract claim.
Good luck to you.
This answer should not be construed to create any attorney-client relationship. Such a relationship can be formed only through the mutual execution of an attorney-client agreement. The answer given is based on the extremely limited facts provided and the proper course of action might change significantly with the introduction of other facts. All who read this answer should not rely on the answer to govern their conduct. Please seek the advice of competent counsel after disclosing all facts to that attorney. This answer is intended for California residents only. The answering party is only licensed to practice in the State of California.