I agree with @Anonymous Mathematician that you should consider applying to American programs anyway. You should write to each one, explaining your situation, and ask if your application can still be considered anyway.
Perhaps I should say that as the Graduate Coordinator of my PhD program in mathematics, I am now getting emails from students saying they didn't take the math subject exam and asking whether can they apply anyway. What I tell these students is that their application can still be considered but will be regarded as incomplete without this exam, so that they should expect to be at a competitive disadvantage. I imagine that this is somewhere in the interior of a continuum of responses that one might receive: some places really might not consider your application and others might not downgrade applicants at all.
In your particular case you have a bit more to say for yourself, namely the part about the accommodations. I don't know what special accommodations you were seeking and it is not my place to ask about it here, but I think that does figure in how sympathetic the committee will be. For instance, if you have severely impaired vision and were asking for larger print and a very well-lighted room, then it sounds bad that the ETS was not able to accommodate that in a timely way. If you were asking for extra time because of text anxiety, the response may be different.
I also want to say that if the special accommodations were in the vein of giving you extra time, then it might have been in your best interests to take the exam anyway. I don't want to get into the issue of how extra time on exams for various conditions is viewed by graduate admissions...so I won't. But you would have given yourself more options by doing so: for instance, perhaps you would have done well anyway.
As others have said, you can certainly apply to math PhD programs in most countries in the world without GRE scores. In fact, I don't know any country outside the US in which most programs require the GRE, although I just checked that a top Canadian program "strongly encourages" it, which is not so far off. (Indeed, in more than half of the academic world, the response to such an inquiry would be "What's a GRE?")
Because advising current and prospective graduate students really is my job now, I am a bit more conscious not to answer too much more than the question was asked. But briefly, I think you should talk to a mentor about your overall grad school strategy and career plans. Learning Japanese and going to graduate school in Japan is a huge undertaking compared to waiting another year to get your application fully in order, especially if your application is otherwise strong. You write:
I am not sure that waiting for a year is a good idea for me because a lot of my application is based off of having accomplished a lot at a relatively young age, and I want to do everything I can at this point to avoid it.
With respect, that doesn't make a lot of sense to me. Programs, even the top programs, are not selecting for precocity; we're selecting for the likelihood and degree of future success. We most certainly do not "normalize" the application by the age of the applicant. I would encourage you to focus on mastery and success in absolute terms, not on getting as far ahead of your age cohort as you can. As I have written elsewhere, in my experience math PhD students who are slightly older than average -- say, starting the program around 24-26 instead of 21-23 -- actually tend to do a bit better, on average. People who get their PhDs in their late 20s need not be "slower" in any way than those who get their PhDs in their mid 20s: rather, they may be more fully using the time and resources they have been allotted.
