In Capri, we stayed at La Scalinatella. If you are looking for a 5-star hotel on Capri, this is it. La Scalinatella is consistently rated in the top three hotels in Italy. In addition to La Scalinatella, there are many more reasonably-priced hotels. Hotel Syrene, which is owned by Best Western, has breathtaking views and is located near the main piazzetta.
In Positano, we stayed at Il San Pietro Di Positano. This hotel was beautiful, but probably more ideal for a honeymoon than a family vacation. The hotel has private beaches, botanical gardens and extensive spa treatments. Because it is secluded from the public beaches, you have to take a (free) hotel shuttle in order to visit downtown Positano. Fortunately, the shuttle runs 24 hours so you don't have to worry about being stranded.
Although Italy is not intentionally vegan-friendly (i.e. there aren't many vegetarian restaurants and most of the restaurant waiters seemed to think I was telling a funny joke when I asked if a particular dish was vegetarian), it is very easy to find a vegan meal while there.
Italians almost always eat a pasta dish for their first course (primi piatti), while the secondi piatti is usually a type of meat. As a result, most restaurants will have a few vegetarian pasta dishes on the menu.
Many Italians are lactose-intolerant, so true Italian food doesn't tend to be smothered in cheese like American Italian food. Unlike Italian restaurants in the US, the restaurants that I visited in Italy never made the assumption that I wanted parmigiano cheese on top of my spaghetti (the waiter would always ask, after bringing out my food).
Salads are also a good option for vegans traveling in Italy. The "salad dressing" of choice in restaurants and most households is Olio e Aceto - olive oil and vinegar (usually balsamic). The salad is typically served with the second course (or in lieu of the second course if you're vegetarian).
The bread served in a restaurant in Italy rarely comes with butter. Instead, you have the option of eating the bread plain or dipping it in olive oil.
There are a few treats that vegans can enjoy throughout Italy. One is latte di mandorla (almond milk), a sweet, refreshing drink that is made from almond paste and served over ice. It is popular in some regions in Southern Italy.
Another vegan-friendly treat is gelato. Gelato comes in two types: milk-based and water-based. The water-based type of gelato is often called sorbetto and is available in a variety of fruit flavors. Sorbetto is the creamiest non-dairy sorbet that you will ever find. Most fruit flavors that you will see at a gelato stand are vegan. The only exception that I have found is coco (coconut), which is sometimes made with dairy.
Capri is an island off the coast of Naples. There are two towns on Capri: Capri and Anacapri. Most hotels and restaurants are in Capri. Capri is also home to one of the most extensive collections of haute couture shops in the world. Anacapri has a quieter, more-subdued feel. In addition to a few restaurants and small shops, Anacapri has a museum (Villa San Michele) and a chairlift to the top of the tallest mountain on Capri. The Villa San Michele was home to Axel Munthe, a doctor, author and early animal rights activist.
While on the island, you can visit the ruins of Emperor Tiberius' many homes. There are also a number of grottos (underwater caves) that you can visit by boat.
Positano is a small town that has been built into the ocean cliffs, which makes for some breath-taking views. There are rocky beaches, small shops selling work by local craftsmen, art galleries and outdoor cafes. The pace is slow. Expect to spend two hours enjoying a leisurely dinner. Positano is a popular destination for American tourists in spite of not having many "tourist" attractions.
Day Trip To Pompeii
It is definitely worth taking a day trip from either Capri or Positano to Pompeii, sight of the famed Roman city that was buried in lava after an eruption by Mt. Vesuvius. Pompeii has extensive grounds that you can explore for a small fee. Over 100 shops and homes have been excavated, some which feature paintings and mosaics that are largely intact.
Food To Avoid (if you're vegan)
Latte - milk
Burro - butter
Uovo - egg (not to be confused with uva - grape)
Formaggio - cheese
Pollo - chicken
Carne - meat
Pesce - fish (not to be confused with pesca - peach)
Prosciutto - ham
L'agnello - lamb