Step 1: Recall the foramen magnum. It is the large opening in the occipital bone at the base of the skull through which the medulla oblongata becomes continuous with the spinal cord.
Step 2: List its contents. The foramen magnum transmits the medulla oblongata and its meninges, the two vertebral arteries, the anterior and posterior spinal arteries, the spinal roots of the accessory nerve (ascending into the skull), and the membrana tectoria and alar ligaments.
Step 3: Test each option against the correct foramen. The internal carotid artery passes through the carotid canal, not the foramen magnum. The sympathetic chain ascends as the cervical sympathetic chain and does not pass through the foramen magnum. The hypoglossal nerve passes through the hypoglossal canal. The vertebral artery enters the cranial cavity through the foramen magnum.
Conclusion: The structure passing through the foramen magnum is the vertebral artery, so the correct answer is option 4.