I see that some consulates have information about this on their websites for instance, but some don't. Are they obligated to provide such accommodations by the law, or it's a merely at-will decision? This question isn't limited by employment matters, but visa applications are included too.
Assuming, for sake of argument without fully analyzing the question, in light of the Vienna Conventions of Consular Relations, that the law applicable to the U.S. Federal Government and not local law applies to U.S. Consulates and Embassies, then the American With Disabilities Act does not apply because the U.S. Federal Government is exempt from that law. Instead, the controlling law would be title V of the Rehabilitation Act of 1973, and in particular Section 504 of that Act which pertains to programs offered by federal agencies. This states:
No otherwise qualified individual with a disability in the United States, as defined in section 705(20) of this title [i.e. title 29 of the United States Code], shall, solely by reason of her or his disability, be excluded from the participation in, be denied the benefits of, or be subjected to discrimination under any program or activity receiving federal financial assistance or under any program or activity conducted by any Executive agency or by the United States Postal Service.
A right for individuals to sue to enforce this requirement is found in Section 505 of the Rehabilitation Act of 1973.
Despite the plain grammatical reading of Section 504 of the Act, 29 U.S.C. § 705(20) defines individual with a disability, but does not clarify the "in the United States" requirement as applied to embassies and consulates.
Taken at face value, this would suggest that Section 504 of the Rehabilitation Act does not apply to embassies and consulates at all since they are not "in the United States."
The U.S. State Department which includes embassies and consulates, like all other U.S. government agencies, has a set of regulations implementing Section 504 in its agency, and those regulations would establish what it is that embassies and consulates must do to accommodate people with disabilities who want to use the services of the U.S. State Department in that context (if anything).
Review of these regulations and further case law research would be necessary to determine if Section 504 of the Rehabilitation Act of 1973 actually applies to embassies and consulates of the United States government.
I will update this answer if I have time for further research, but I would hope that this answer at least focuses any further research by the person asking the question or anyone else seeking to answer it.
US Consulates (and Embassies) in foreign nations will be subject to applicable US law concerning disability accommodations. This generally refers to the Americans with Disabilities Act and would apply to foreign nationals who work at the consulate. This law is a matter of public record, so the consulate might or might not see fit to mention its participation in following the law.
Employment law in a foreign embassy may be a topic of contention between nations. The Vienna Conventions on Consular Relations which govern consular activities in general state that the grounds are considered inviolable, that they are not subject to taxation, and that members of the consulate have immunity from certain prosecution. It does not include that the organization itself is exempt from local law, although in practice enforcement of such actions would turn into diplomatic issues as much as legal ones. So accommodation law in the local nation might be an issue. We do find that for a job at the US Embassy in France, one must be eligible to work in France to apply to work there (this does not apply to "diplomats"). I believe that in practice the guideline is that the consulate does not have to follow local law, but the local employee does. I also imagine huge buildings filled with government employees managing, deciding, and getting guidance from government lawyers on these types of things.