The question is about identifying which organization is responsible for the delimitation of the Exclusive Economic Zone (EEZ) as stipulated by the United Nations Convention on the Law of the Sea (UNCLOS III). Let's analyze each option to find the correct answer:

1. International Seabed Authority: This organization is mainly responsible for regulating mineral-related activities in the international seabed area, which is the part of the sea beyond national jurisdiction. It is not responsible for delimiting the EEZ.
2. International Tribunal for the Law of the Sea: This tribunal settles disputes and gives advisory opinions regarding maritime law but is not directly involved in delimiting the EEZ.
3. International Boundaries Commission: No such commission exists under UNCLOS for delimiting EEZ. This appears to be a distractor.
4. United Nations Commission for Limits to the Continental Shelf: This commission is indeed responsible for facilitating the establishment of the outer limits of the continental shelf beyond 200 nautical miles. It is closely related to the delimitation process within maritime boundaries stipulated under UNCLOS.

After evaluating the options, the correct answer is the United Nations Commission for Limits to the Continental Shelf, which plays a critical role in delineating maritime boundaries and defining the limits of a country's EEZ.