What address to use from the US to request tickets for General Audience with the Pope?

It would probably be faster to send your request by fax or email to the U.S. Bishops’ Office for Visitors to the Vatican at visitorsoffice@pnac.org (details on schedule and how to make a request here). If you're staying in a hotel with concierge services, this may be something they can arrange for you.

If you do want to send a letter by mail, I suspect USPS just screwed this one up. While it's not correct, putting "Rome, Italy" on the end of the address may be enough to clue them into routing it in the general direction of Europe. "Vatican City, Europe" on a new line at the end of the address may work too.

But I suspect that fax or email will be easier and faster.

An interesting fact: The primary official language for international postal services is French. Anything properly addressed in French cannot be refused. English was also added as an official language but only as late as in 1994. These days there are systems that are capable to recognize many different languages but they might not be working everywhere so limiting yourself to writing (at least) country name in French or English is a good idea.

Also the last line should be just the name of a country, capitalised. Some of the countries require to capitalise all letters, not just the country.

On the UPU page you can find examples of addressing in accordance to a specific country's format. Note, the addresses are stripped off of the country, that should be, as already mentioned, placed in the last line.

Finally it's always a good idea to go to your local postal office and ask how to properly address your mail. Let me tell an anecdote here. My father, located in Lodz, the third largest city in Poland, had to send some business mail to the British Virgin Islands about 10 years ago. So he wrote the address and went to the post office to send it. The lady at the counter deck looked at the address and said:

There is no such country as British Virgin Islands. There are Easter Islands, Islands of Cape Verde, but no British Virgin Islands (in Polish all those names contain the word "Islands").

So my father went back to the office, and discovered that French is the main language. So he went once again to the post office with a mail readdressed in French, but he got refused again. He asked for a manager, explained everything and the manager eventually accepted the postage claiming she will make sure it will leave the post office with no further obstruction.

After a few days, the mail was returned with a Warsaw postal stamp (Polish capital city) and annotation "There is no such country". My father's conclusion was "the British Virgin Islands might be a tax paradise, because it is not possible to send any debt reminders there".

If you don't mind another answer, what I think is happening is that your address is confusing the automated systems.

Instead of reading the line as "00120 Vatican City State" and correctly routing it to Vatican City in Rome, the software is reading as:

Street address: "00120 Vatican"

City and state: "City State" (missing zip code)

And since there's no such city called "City" and no such US state as "State", the letter was returned. So the previously answer of adding "Vatican City" on a separate line as a separate country is correct.

I would also recommend taking your letter to an office (US Post Office, UPS Store, and so on) so someone can type the address into their computer to get the correct postage amount and to route the letter so it arrives correctly.