If a person has health reasons that make sleeping late (say, 3am-11am) a necessity, can such a person appropriately hold monastic vows in the Geluk tradition of Tibetan Buddhism? Are there certain sleeping restrictions mentioned in the vows?
Yes, such a person can appropriately hold monastic vows in the Geluk tradition of Tibetan Buddhism. There are sleeping restrictions in the Vinaya, but not regarding the time or length of sleep. Although Vinaya states that sleep is an impediment to the three trainings, in any case over-sleeping (are 8 hours of sleep over-sleeping? this is open to interpretation) would not constitute a downfall since it is not the breaking of a vow.
However, sleeping at such hours and for so long would be seen as inappropriate by many. In addition, a monk could hardly stay in a monastery or in a center (even in the West) with such a schedule, because there are (1) monastery-schedules, and (2) rules that are not Vinaya but institutional. It is not rare to have pujas in the morning, etc. Of course, Western institutions are not as strict as the Tibetan ones (for Tibetan have to wake up at 5:30 am, and if they do not, they are fined by the disciplinarian) but still.
Most of the western sangha are relatively wealthy elderly women who live in their houses, enjoying various floors, living with their cat, great gardens and the like, on their own, and not all taken that seriously by monks.
I suggest you read Lama Yeshe's Advice for monks and nuns.
When you are true to the practice of Dhamma, with time you tend to sleep less and less due to the ‘tranquility’ or 'calm abiding' that you experience in your day to day life. So your late sleep schedule is not a given. It is going to change to the better. In practice you learn to push yourself to alter negative habits. When it comes to sleep, at times you may back off from this kind of pushing a little bit until you find what is the number of hours of sleep is right for you. For each of us, this is going to be different as to how much sleep we need. This will inform you as to how much you can push yourself in the practice. The important principle is you learn to be responsible for yourself—because this is how discernment is developed. Too much sleep is a hindrances to the pursuit of this path. The Dhamma word for this is “thina middha”.
Once when the Venerable Maha Moggallana, one of the Buddha's two chief disciples, was meditating in the forest, thina middha arose. His mind shrank and withered, as unworkable as a piece of butter that hardens in the cold. At this point Supreme Buddha looked into the Venerable Maha Moggallana's mind. Seeing his plight, he approached and said, "My son Maha Moggallana, are you drowsy, are you sleepy, are you nodding?" The elder replied, "Yes, Lord, I am nodding." He was frank and candid in his reply. the Buddha said, "Listen, my son, I will now teach you eight techniques of overcoming sloth and torpor." Some of the eight Ways to Stay Awake are… first is to change one's attitude. When torpor attacks, one may be tempted to surrender to thoughts like, "I'm so sleepy. It's not doing me any good just to sit here in a daze. May be I'll lie down for a minute and gather my energy." As long as you entertain such thoughts, the mental state of sloth and torpor will be encouraged to remain. If, on the other hand, one states decisively, "I'll sit through this sloth and torpor, and if it recurs I still won't give in to it," this is what the Buddha meant by changing one's attitude.