It depends on how flexible you are and if you're willing to take what you can get at the last minute. Because July is Peak Season there is stiff competition for the best rooms in the best locations but there is always _something_ available as long as you understand that it might not be in the best location, or it might be more expensive than you would normally pay. You won't know which until you get where you're going and give it a try.

One suggestion I would make to you is to compile a list of possible hotels on the islands you've chosen to visit and call ahead from wherever you are when it's time to change islands to see what you might be able to reserve in advance that very day. Also, when you arrive look around for room-finder businesses. They're usually close to the ferry dock.

The first week of July shouldn't be difficult to find accommodation on the islands, even though this is the beginning of the High Season. It will become much more difficult after the 20th of July.

If you are very picky, and want to have a room on a specific beach with sea view ( and not ocean view , as i always read on this forum... since the Mediterranean is not an ocean...) then booking in advance is recommended.

What always worked out for me, was having a look at www.booking.com, at what is available. You will be surprised by the big number of hotels, pensions etc, that are available even on same day basis.
You can then either book it online, or simply go to the hotel that appeals to you and make a direct booking.

On most islands, as brotherlee said, there will be people waiting upon your arrival, with folders showing photos of their accommodation. Just be very careful with their description... 5 min from the harbour can be 30 min walking, on the top of a hill... ( happened to me ages ago.....)

As Clausar mentioned above, I also suggest www.booking.com .
Choose the hotels offering the option 'free cancellation'.You can book three hotels for the same date and cancel two of them one or two days before your arrival---without penalty.
In this way you will have more time to decide.
Have a nice time in Greece

It really depends upon which island. You are wise to book in advance for Santorini, because many travellers to that island seem to book well ahead, especially for hotels on the caldera side. I wouldn't turn up on a small island like Folegandros in July without a reservation, because accommodation is very limited. Paros and Naxos should be safe enough to wing it in early July.

You have gotten some good advice above. Learn as much about the geography of each island so you know specifically where you would like to stay, e.g. Parikia or Naoussa on Paros, Naxos Town or Agios Prokopios on Naxos. When I am island hopping without reservations I like to make a shortlist of hotels on each and print out a page from their website to bring with me. Then you can either call ahead, as brotherleelove suggests, or look at the portfolios of the hotel owners waiting at the dock. A phone with a Greek SIM would be helpful for calls ahead. Some of the best deals are from small hotels that don't even have an internet presence, and they are the ones usually waiting at the dock. Using booking.com will guarantee you a room if you prefer that.