First Time in Rome — VRBO Apartment Recommendations for a Family of 3?

See this thread —- https://community.ricksteves.com/travel-forum/italy-reviews/favorite-rome-apartments

Any budget limitation?

Our budget is up to $4,000 USD for 5 or 6 days.

I stayed in a VRBO apartment called NAVONA, CAMPO DI FIORI - Penthouse Terraced in November 2022. I loved the apartment and the location. it meets all your requirements.

I left a review, which you can find if you scroll down to the 2022 reviews. I will add that my husband broke one of the chairs on the balcony. The canvas split. (He weighs about 190, nothing excessive.) I notified the owner at once and offered to pay for it, but I noted that the chairs seemed to be quite old and were starting to fall apart. He asked me to send pictures, and after I did, he said we did not have to pay.

If you are open to Airbnb, search for Casa Pantheon, hosted by Tina. We had a great experience in 2017, and the reviews are still good.

My family of three stayed at an AirBnB called Home in Rome that we really liked. The apartment was situated just off Piazza di S. Cosimato in Trastevere. While a lesser known, maybe unknown, piazza the location had many things going for it. There is a playground in one corner of the piazza that my young daughter enjoyed. There is a pretty good market two days a week. In summers the piazza hosts a film festival. And it is quite near the Juniculum hill with its Fontana dell'Acqua Paola. Also near the Basilica di Santa Maria, an atmospheric church.
The host was born and raised in Rome. It is only one bedroom but there is a sofa that unfolds into a bed in a corner of the spacious kitchen. I think you'd be able to find the listing on VRBO as well.

We stayed in an Incentrum apartment and liked it a lot. Our building was located between the Piazza del Popolo and the Spanish Steps, and the other building is not far from there. Note that there are 2 buildings, and one has an elevator and the other does not, if that matters. (There were a lot of steps to our apartment on the 5th floor (6th floor in North America).) Anyway, there are a variety of apartment configurations available.

https://incentrum.it/en/home-page/

I don't know your financial situation but just so you know, your budget is VERY high. Your must-haves are pretty basic, so I would not call them "lofty" unless you were being sarcastic. You can easily get something for under $1000 for 5~6 days.

I don't know if you are fixated on VRBO or if you are just using it as a generic term for vacation rentals from an owner. My personal experience is that Airbnb is much more popular in most foreign countries.

I booked an apartment for my wife and I for Oct 2027 (mirror image of your April date as shoulder seasons) for a little over US $100 a night for a week, within walking distance of Vatican. It has 2 bedrooms and would have fit your family perfectly.

I cannot give you a personal recommendation since we haven't gone there yet. And if you get recommendations from other people, since it's not a hotel but a single rental unit, the chance of that specific place is avaiable for your specific date is not guaranteed. I think you would be better off choosing 1 or 2 neighborhoods (based on general reviews from Rick Steves or an AI bot), then do a search on Airbnb (or VRBO if you prefer) and filter the options. I have found that anything 4.7 * or higher is usually really good. And once you narrow down to 4 or 5 places, spend some time to actually read each individual review, as you can glean a lot of useful information about the place and the neighborhood from the reviews.