There are several layers to this. I want to make sure that I understand what you are trying to do. There is something informal. For example, once I was in Naples with a couple, and we struck up a long conversation with a nun, purely by accident. The nun lived in Rome. She said she was going to get a papal blessing of the marriage of the couple, and took down the name and address. Sure enough, a few months later they received a letter from the Prefecture of the Vatican that the Pope had blessed their marriage. I don't know if he just said a short prayer and sign of the cross over a stack of letters and rosaries, but the nun did follow through, and there is documentation of the blessing.
The next step is more formal. It is to get a Papal Blessing by Parchment. You get a very nice, framable parchment indicating that the Pope has specifically blessed you. This is done by mail. They sell different levels of Parchment quality, I think ranging from around $15 to $30. It basically just needs your name and address, method of payment, a description of the occasion for which you are seeking the blessing. It could be anything from a baptism, to a marriage, to a marriage anniversary, or a blessing for your whole family.
Then the next thing, which is what I think you want, is to meet the Pope and get a personal blessing as newlyweds. This is going to be hit or miss. About a year and a half ago I brought a lady to Rome with me. She was not very experienced at travel, but was very Catholic, and dreamed about being able to see the Pope, but figured it's impossible.
Actually, it's not that hard to see the Pope. He holds a Mass every Wednesday morning at a small venue in the Vatican. You need tickets, but they are free. You mail in a request a few months before, and you will receive confirmation. If I remember correctly, we picked the ticket up at the Vatican the day before the Mass. I didn't want to go, but I took her there to get her ticket. How close you get to the Pope depends on two things. The first is the season of the year, which determines how many people are visiting Rome. I think we went in January. The second is, how early you arrive. That's dependent upon the time of the year. In June, I'd think about arriving at least an hour and a half early. The lady I took was about 20 feet from the Pope. She sent pictures which I have somewhere on my hard drive, but cannot easily find.
Now, what I think you are really talking about is that the Church honors newlyweds, and during that Wednesday Mass there is a special roped off area for Sposi Novelli, or newlyweds. I think you are looking for a ticket to get into that cordoned off area. To get a ticket to sit in that section mainly relies on your priest. He makes the request, provides the date of the wedding, name of the church, and where the wedding happened. Women usually wear their wedding dress, or buy something white that looks like a wedding dress when they get to Rome. Men dress up too, of course. Bring your marriage certificate.
Since the Vatican in general doesn't deal with emails, only letters and faxes, the best way for you to get help arranging this is by emailing
[email protected]. This is a seminary in the Vatican that in addition to its seminarian duties, is the Vatican's Visitors Office for Visitors from the USA. They take emails, so they can guide you through this more efficiently that sending a letter to the Vatican, then waiting for two months.
There are a few possible disappointments you should keep in mind. Newlyweds are cordoned off, and get to sit much closer to the Pope during the Mass. After the Mass, he goes over there and gives a blessing. Since there are a lot of June weddings, so that when he comes over, you still might not get close enough for example, to get a handshake. They do have a way of parading you buy him with a professional photographer. Then they send an email to you with proofs of photos of the event, and you can choose which ones you want to buy, for example if there happens to be one where you were able to get right next to the Pope.
And here is the biggest thing to know. They do this every Wednesday, but only if the Pope is in town. He does travel around as part of his duties, so you have to go to the Vatican website and check his schedule. The problem is, they don't publish the Pope's schedule until about 2-3 months in advance. Also, I don't remember if it's two months or three months, but after that time you are no longer eligible because you are no longer newlywed. Sometimes, Popes are not in Rome during the summer. Popes have a castle outside the city where it's cooler, but I hear that Papa Francesco has not yet used it, and most Popes don't start using it to escape the summer heat until July, so you should be OK. Just hope he is not traveling on some mission to somewhere on your dates.
Send an email to the Vatican Visitors Office at the email above. It's run by nuns. All of the nuns are from the USA. And it is a free service. They will coordinate for you.