Looking for recommendations for accommodations in Mykonos and Santorini at the end may. We are a family of 5, so we need an apartment or a short term rental. We will spend 3 nights in each place and would prefer something with a view

To help you get recommendations, give a nightly budget.

Mykonos: Petasos Beach Resort

Early May 2022 we spent three nights at Olympic Villas just outside Oia on Santorini, we were a group of seven and booked three appointments, one of which was a cave room. The apartments have a small kitchenette and a patio with gorgeous caldera views, it’s about 10 minutes to walk into Oia. They have some 2 bedroom apartments, but all have only one bathroom. At that time we paid about 250/night including breakfast, which is quite reasonable for Santorini.

Oia is very crowded with tourists during the day, but we rented cars and drove around the island, Oia was stunning in the mornings and evenings once the crowds left. Incidentally, there’s a car rental agency right across from the hotel, you can also park tcarcthere at night, it was most convenient.

Santorini: Check Ifestio Villas. They offer a 3-bedroom apartment.

Have you considered staying on Paros and/or Naxos for a more real Greek, less touristy experience? We couldn’t wait to leave Santorini and the hordes of tourists.

We really enjoyed our stay at Blue Dolphin Suites in Firestefani on Santorini. We had a lovely room with a balcony overlooking the caldera. The rate was $230 per night. Several mornings we had breakfast on our balcony. The cost is 25 euros for 2 people, and it was amazing! Pancakes with fruit sauce, eggs, sausages, Greek yogurt, pastries, bread, coffee and juice.
https://www.bluedolphins.gr/

I would recommend Myconian Collections Hotels and Caldera Collection Hotels in Santorini if you have the budget.
For value for money in end of May that both of the island would be crowded I would recommend you stay in downtown Mykonos (Chora) and downtown Santorini (Fira) to avoid local transportation and taxis.
Or else if you are thinking staying on the beach instead, Elia (Mykonos) and Kamari (Santorini) are the recommended locations.