The only people who hold citizenship there right now are the Pope, Cardinals, other clergymen, and Swiss guardsmen. None of whom are either fit to be Olympics or have the time to train for the Olympics. You would need to have a larger base of citizenship than what they have now. One way could be to change the laws of what constitutes citizenship in the Vatican such as including family members of the clergy or include people born in the Vatican to be citizens. This could be relatively easy, but still the pool would be very low. But you could end up with a Olympic Athlete.
Another way, albeit more difficult to do with a POD past 1900 but would increase the citizen pool, would be to increase the size of the Vatican. I can't see Italy willingly give up large swathes of Rome to the Church after unification. Maybe Italy is broken up after WWII, and the Papal States become their own thing and compete in the Olympics?