I am a F1 student from Chile. My wife is in F2 status (also from Chile).

I file federal taxes as nonresident (hence I can not file jointly with my wife) using my SSN.
Since my wife has no income and only files form 8843 and can not be claimed as dependent (there is no tax treaty with Chile) she is not eligible for ITIN (of course, she is not eligible for SSN either). In fact she can not even apply since for applying you need to file W7 along with some federal tax form that requires her to have an ITIN, justifying her application.

For federal taxes this is fine (you CAN file only 8843 without ITIN), but for MA state taxes they ask for either SSN or ITIN. (We file state taxes as nonresident married filing jointly)

What should we do?
Can we file MA form 1-NR/PY without ITIN for her?
Or can she apply for ITIN in another way? (without a federal tax form that requires it)

You CANNOT file the Massachusetts return jointly, because you did NOT file that way on the federal return and, as you noted, your wife has NO ITIN.

You have to file the Massachusetts return MARRIED FILING SEPARATELY, insert your wife's name and put "NRA" (for Non-Resident Alien) where her SSN would go. This Massachusetts state return MUSt be mailed.