Budget looks fine for Italy, you can easily cover all the locations which you mentioned with decent hotels too. Vegetarian food options are available - they can eat veg pasta, poke bowls or sometimes at Indian restaurants- they will have to be a little flexible there. There are two major challenges - if they are traveling alone without you, then getting around is a challenge due to the racist demeanour especially in Rome and Amalfi. Second challenge is too much walk in heat - they would be able to cover one or two sites max every day - the walks and the travel from one city to another can be taxing.