When it comes to the big standardized admission tests for (mostly) US graduate programs, such as the GRE and TOEFL tests, typically only Qwerty is allowed. See below for their policies. That said, they do make some accommodations for disabilities. Being a Dvorak typist does not count by itself, but if you have carpal tunnel it can't hurt to ask for special accommodations ahead of time. By the way, that link leads to a Dvorak community on Reddit. You might get better answers there.
For admissions and placement tests administered by individual universities, you would have to ask them. No universal rules there. For computer based exams and similar once admitted to a program, I think your chances are quite good. Few people would care about what keyboard layout you use, and even if they would, student disability offices tend to be quite flexible.
GRE
The GRE General Test uses a standard English-language (QWERTY) computer keyboard. It takes its name from the first 6 letters in the third row of the keyboard. If you haven't used this kind of keyboard before, practice on one before test day to become familiar with it.
The GRE Bulletin Supplement for Test Takers with Disabilities or Health-related Needs does mention some keyboard-related accommodations (e.g. ergonomic keyboard), but I don't see anything about layout. Unfortunately, I don't know how likely they are to make other accommodations.
TOEFL
The TOEFL iBT test uses a standard English-language (QWERTY) computer keyboard. It takes its name from the first 6 letters in the third row of the keyboard. If you haven't used this kind of keyboard before, practice on one before test day to become familiar with it. In some countries, the common keyboard used is configured to QWERTY and a template is provided to each test taker to help with locating the few keys that are in a different location.