"I can only work in one organization." is awkward English and should not be accepted. "I can only work at one organization." OR "I can only work for one organization." should both be accepted, but not "in". Why? The preposition "in" is typically used for fields or specific physical locations, example: I work in sales. I work in computers. I work in the medical field. I work in Houston. I work in town. I work in a small building. I would never work in an organization, but I may work in a department of the organization. I work in shipping at Safeway.
To the extent that the course is about learning Hebrew, and not English, maybe they all should be accepted. The canonical correct answer should be "for".
You are correct that "for" is the best preposition. But I would also point out that only should really be in a different position. It should be, "I can work for only one organization." Or: " I can work for just one organization." Someone else posted also, "I can work for one organization only." Someone replied that they did'nt like that, but it sounds fine to me. Depends on what you're emphasizing.