We'll have 3 days each in Naxos, Santorini abnd Mykonos in July (and 3 in Athens). My wife has a bad knee, and also can't operate a scooter. I'm thinking that one day will be exploring the main towns (on foot), but how should we get to the other Island highlights? Do they rent two-seater scooters?

Use the local buses or take a taxi - they're cheap in Greece.

You can rent a car easily on Santorini. A couple of boys I met rented a little smart car for 30 euro a day. I hear you can get a car for even cheaper. There are car and scooter hire everywhere. If you'd like, I can get the name of the place they rented from.

Although in July, I am not sure if the traffic would be bad. It's pretty easy going around the island now. I was told that in the summer, the buses can get intolerably crowded.

2-seater scooters are no problem, but because of the way the villages ovelooking the caldera are laid out, it will be difficult for someone with a bad knee to walk around without discomfort.

There are good bus systems on the islands now as well.

You might regret the scooter idea, it will be crowded in july and if your wife already has a bad knee, the thought of swerving around crazy holidaymakers on scooters might not be her idea of fun.
The taxis are reasonably priced but the local buses are a lot of fun.