I agree with Janet about choosing Sifnos over Naxos, especially as Sifnos becomes more and more popular every year, go there before it becomes overrun with tourists from June to September just like Naxos or Paros.
See the Sifnos website:
https://sifnos.e-sifnos.com/
I also agree about Aegean Thesaurus Travel, a travel agency in Sifnos which has existed for a very long time and which found accommodation and car/scooter rentals for visitors before the internet era.
https://www.thesaurus.gr/
That being said, you can also use the usual services like booking.com or Airbnb to find accommodation.
If you don't want to go through an intermediary see "rooms in Sifnos" a locally managed website where you can contact the owners directly
https://www.roomsinsifnos.com/
Almost all the beaches are suitable for young children like those of Platis Gialos and Vathy mentioned by Janet, I would also add Faros which is less busy but in the same area.
I do not recommend Kamares, the port which, although having a beach which may be suitable, is in the North and is in my opinion a place in Sifnos for people who do not want to discover Sifnos.
Sifnos is also famous for its gastronomy, which is also why as foodies we will probably be there next September for the Cycladic Gastronomy Festival :))
How likely are ferries between Piraeus and Sifnos to be affected by
weather in mid-June?
Although bad weather preventing ferries from sailing is infrequent there is always a risk, whether in June or any other month.
Anyway you will only know the weather forecast 3 or 4 days before the trip. So no need to worry about it 15 months in advance.