Does the Vatican City participate in the Olympics?

Not under a Vatican City flag. There are 202 nations/independent territories/ commonwealths/ protectorates and geographical areas that have been issued 3-letter abbreviations (trigram) by the International Olympic Committee. The trigrams are issued to countries with National Olympic Committees (NOCs). NOCs are in place to ensure the nations best athletes attend the Olympics. Vatican City does not have a NOC and so its athletes cannot compete as Vatican City athletes.