I'm not 100% clear whether you want to buy a house for you (and/or your relatives) to move there permanently or just as a vacation home.
If you (and/or your relatives) plan to move there permanently, and you (and/or your relatives) are not EU citizens, the only possibility is to apply for a long term visa called ELECTIVE RESIDENCY VISA, a visa that the Italian government grants to people who plan to retire in Italy. It's not a work visa, since with that visa one is not allowed to work. I'm giving you below the page of the Italian Consulate in Los Angeles, which I believe has jurisdiction over Arizona, to get more information.
https://conslosangeles.esteri.it/en/servizi-consolari-e-visti/servizi-per-il-cittadino-straniero/visti/visa-type/elective-residency-visa/
If the plan is to retire in Italy, be known that Italy has much higher income taxes than the US. Their top income tax rate is 43%, which however hits incomes above 50,000€. The income tax rate below that one is 35%, and hits income above 28,000€, so as you can see the tax rates are high and hits incomes at very low level. Like all European countries, Italy likes to soak the middle class and also the poor since there isn't even a standard deduction. Multimillionaires actually pay less taxes in Italy than in the US, since foreign multimillionaires moving to Italy can choose to pay a flat amount of 200,000€ regardless of how many millions they make income, so basically for any yearly income above 1 million or so, the amount to pay is 200k euro, and at 20% (or less) it is probably a good deal for those who qualify.
Since I gather that is not your situation, then, not being rich, you will be taxed heavily. The bilateral treaty on taxation between Italy and the US provides that Public Pensions continue to be taxed by the US, unless you are an Italian citizen, while private pensions (like 401k) are taxed by Italy. There are also wealth taxes on assets held outside of Italy, in case you or your relatives decide to keep bank accounts and real estate in the US.
Given the above tax considerations, you should also know that if you move to certain regions and establish residency in one municipality with less than 20,000 population in those specific regions, you will be exempt from Italian taxes for 10 years. The specific regions that qualify are basically the southern regions and the islands, namely: Abruzzo, Molise, Campania, Puglia, Basilicata, Calabria, Sicily, Sardinia. So if you want to save taxes, it would be a good idea to move to one of the aforementioned regions, and of course you must select a municipality with less than 20,000 population.
There is a Canadian lady in this forum who did just that and moved to Abruzzo, not sure which small town. Her name is Nelly and maybe she can give you some advice on Abruzzo, since you are considering that region. Once you zero in a specific town, you can see the population on wikipedia so that you make sure the municipality has less than 20,000 pop. Some of the towns in Abruzzo which qualify are within one hour from Rome, if you want to be close enough to a large city and airport.
For real estate websites I recommend www.immobiliare.it or www.idealista.it. Both websites have an English language option.