Sorrento , Capri and Positano for kids

Capri is pretty much an adult-centered destination but the page for kid-friendly activities from the website might be helpful:

http://www.capri.com/en/e/capri-with-kids

I believe the playground mentioned in Anacapri is marked as "Giardini Pubblici" on google maps. They don't recommend specific restaurants but kids are almost universally accepted anywhere in Italy so whatever looks good from the website's list. I'd go for easy food like pizza, or you might pick up some takeaway from Al Buco (Anacapri) and have a picnic?

http://www.capri.com/en/c/al-buco-rosticceria

Or pre-order a picnic basket of goodies from Ristorante Michel’angelo (Capri) and they'll even give you some suggestions for less-visited places to enjoy it and rent you a blanket for a small fee.

http://www.caprimichelangelo.com/picnics.html

One comment about the chairlift up Mt Solaro (mentioned on Capri.com)? I've done it and it was fun but I wouldn't do it with a small child on my lap and the website doesn't recommend it either. And skip the Blue Grotto. The most fun thing to do on the island is just to explore the tiny "streets" and narrow passageways and enjoy the view!

Capri did not strike me at all like a kid destination - more like luxury shopping, dining, strolling, and hiking. There are lots of things to appeal to adults, but not so much small kids. There are some small parks in Capri - Gardens of Augustus, but hiking anywhere may (or may not) be too much for some 5 year olds (I recall some steep sections but you'll just have to play by ear). The beauty of Capri is to see all the different vantage points on the island, as well as views, which may require hiking a bit on uneven surfaces and ascending and descending.
https://www.tripadvisor.com/Attraction_Review-g187783-d2489130-Reviews-Gardens_of_Augustus-Capri_Island_of_Capri_Province_of_Naples_Campania.html

I believe there is a playground at Parco Giochi Don Luigi Verde in Sorrento, very near Grand Hotel Ambasciatori. For a fabulous view, Villa Comunale is wonderful: bring gelati! We especially enjoyed this one at sunset. :O)

In Italy, kids play in the town squares and on the beaches...there are lot's of open squares in Capri and Anacapri near churches....personally, I would get a baby sitter from the wedding party for the day and enjoy Capri on your own.

In Italy people take children to restaurants, because they are generally well behaved...but tourist areas are full of pizza, gelato, pastries so I am not sure what you mean about restaurants for a day trip....the time will go quickly, pick up something and eat with a view, then keep going...you will only have until about 5 or 6 pm until the ferries go back to Positano. If anything get down to Marina Grande at least 1 hour ahead of the departure time and relax in a cafe right there with a view of the port.