It's like "use" and "utilize". Either is correct.
http://dictionary.reference.com/browse/acclimate. ac⋅cli⋅mate
/ˈækləˌmeɪt, əˈklaɪmɪt/ Show Spelled Pronunciation [ak-luh-meyt, uh-klahy-mit] Show IPA
Use acclimate in a Sentence
–verb (used with object), verb (used without object), -mat⋅ed, -mat⋅ing.
to accustom or become accustomed to a new climate or environment; adapt.
http://dictionary.reference.com/browse/acclimatize. c⋅cli⋅ma⋅tize
/əˈklaɪməˌtaɪz/ Show Spelled Pronunciation [uh-klahy-muh-tahyz] Show IPA
Use acclimatize in a Sentence
–verb (used with object), verb (used without object), -tized, -tiz⋅ing.
to acclimate.