Can I apply to take the exam if I am not a US citizen?

In general, the answer is “no,” however there are exceptions. Applicants who are not United States citizens and do not reside in the U.S. are not eligible for registration except if the individual is registered to practice before the patent office in their country of residence AND if their country has a reciprocity agreement with the U.S. (Canada is the only country with such an agreement at this time).

There are some other exceptions:

An alien residing in the United States may apply to take the registration examination. To be admitted to the examination, an applicant must establish that recognition is consistent with the capacity of employment authorized by the United States Citizenship and Immigration Services (USCIS). The evidence must include a copy of both sides of any work or training authorization and copies of all documents submitted to and received from the USCIS regarding admission to the United States and a copy of any documentation submitted to the U.S. Department of Labor. A qualifying non-immigrant alien within the scope of 8 CFR § 274a.12(b) or (c) is not registered upon passing the examination. Such an applicant will be given limited recognition under 37 CFR § 11.9(b)if recognition is consistent with the capacity of employment or training authorized by the USCIS. Documentation establishing an applicant’s qualification to receive limited recognition must be submitted with the applicant's application. Qualifying documentation would show that the USCIS has authorized the applicant to be employed or trained in the capacity of representing patent applicants before the USPTO by preparing and prosecuting their patent applications. Any USCIS approval pending at that time will result in the applicant's application being denied admission to the examination.