What other word or words can be used to replace "fit in"? E.g. "I believe I will fit in with your organization."
Assimilate - To become or allow something to become a part of a country or community rather than remaining in separate groups. Eg. New arrivals find it hard to assimilate.
I'll offer a couple of suggestions that would work well in the context you gave (all definitions chosen for the appropriate context from dictionary.com):
to match, coordinate, or interlock: The two versions of the story don't mesh.
to bring together or incorporate (parts) into a whole.
to make up, combine, or complete to produce a whole or a larger unit, as parts do.
to unite or combine.
to establish a social or sympathetic relationship with a person or thing: two sisters unable to relate to each other.
Integrate would probably be my chosen verb for describing the ability to work well with co-workers in order to complete a project in a professional setting. When talking about how well people relate to one another, I believe it invokes a more personal connotation.
Conform - "behave according to socially acceptable conventions or standards". See https://www.google.com/search?q=conform ; "fit in" is listed as a synonym