Based on your description, the halachic/legal status appears to be that the foundation rents the apartment from the campus and rents it out (with the campus’ permission) to you in the form of part of your compensation package.
The first question we must discuss is whether one who rents is allowed to sublet. One who rents a movable object is not allowed to give it to someone else without permission from the owner.[1] However, when it comes to real estate, the renter is generally permitted to sublet.[2] The Rambam explains that since the renter has full rights to use a rented item in any non-damaging way that he wants during the time of the rental, there has to be a good reason to disallow him to rent it out to someone else. Regarding movable objects, such a reason exists; there is a concern that something will happen to the object, and the owner has the right to not trust the second person’s honesty about telling what transpired. This is less of a concern regarding real estate. The Rambam makes a practical, common-sense distinction – one may not sublet to a family that is larger than the renter’s family.[3] The Rama[4] adds that one may sublet only to an upstanding person.
In matters of this type, the local minhag supersedes classical Halacha.[5] Unless there are strong indications otherwise, we assume that a local (in this case, Israeli) law sets the standard. Clause 22 of the Law of Renting and Borrowing states that one must ask permission from the owner before subletting, but if the owner objects on unreasonable grounds, his objections may be ignored.
You should not sublet the apartment without discussing the matter with at least one of the parties. Both the law/minhag and probably Halacha mandate that you give the owners (the campus) the opportunity to express any objections, which might include matters you did not consider. Furthermore, the foundation cannot give you more rights than they have themselves, and since it is common for a rental contract to disallow subletting without permission, you need to ascertain what the foundation’s agreement with the campus was.
Assuming that you receive permission from the campus, the more difficult question is whether you have to receive permission from the foundation. The foundation cannot, unless previously stipulated, force you to let them rent out the apartment during your vacation and keep the money themselves, as it was put at your disposal for the entire period of your stay. However, what if they say, “If you want to sublet, we want (some of) the money”? Since the potential problem of subletting is a matter that affects mainly the owner (his property could get damaged), if the campus does not have concerns, the foundation can probably not raise issues, unless they are renting it long-term and will want it in good condition for the use of a different worker after you.
The question relates more to your compensation package. The foundation might be able to claim that the apartment rental was included in your salary only as necessary for you to have a place on campus in which to live and out of which to work. They may not have meant to include it as an additional money-maker for you. We cannot tell for sure who would be right if such a claim were made without hearing both sides’ claims. We also do not know if there could be any sensitivities regarding the relationship between the campus and the foundation. Therefore, even if for no other reasons than menschlichkeit and to maintain good favor in your employer’s eyes, we feel that you should inform both the campus and the foundation of your intention to sublet and ask if it is acceptable to them.[6] (You do not have to suggest sharing the proceeds.) If there is an objection, hopefully you can work it out, or you can ask the question again based on the new situation.