First of all, you need not be Catholic or even Christian. The purpose of your stay need not be religious or spiritual. Most importantly, the places included on the website are not necessarily what we consider Catholic structures e.g., monasteries and convents. Some are family-run farms, others look just like hotels. All have been approved and have some connection to the CEI (Conferenza Episcopale Italiana), the organization of Italian bishops.
Now. There are a total of nearly 316,000 beds available in these facilities, which are wildly different and cover the entire country, from the Alps to Sicily. The greatest number are in Lazio, the area of which Rome is the capital, and Veneto, the area of which Venice is the capital. There are places with wifi, pools, air conditioning, and restaurants. The prices are usually quite competitive.
Here is just one example. Above you see one of the many offerings, a family-run house in Caprese Michelangelo, near Arezzo in Tuscany. It has a pool and a restaurant. Prices for singles are 30-35 euros, depending on the season.
The portal is called Ospitalita' religiosa and can be found here. On the right side of the home page you will find access to the various translated versions, including English of course. Each entry gives detailed information on the specific facility. An excellent search function allows you to refine your search not only by location but by other fields such as amenities. You will not be able to make reservations from the site, but each entry has a form you may send to the place you are interested in.