Yes. I have been on Long Island and once went to Jones Beach. That was many years ago however. I know the western part is very densely populated away from the beach. That means there's more chances for rentals. I just looked at a map online and Mastic and Mastic Beach have bicycle lanes through the town to the beach. I don't know the rental market but in my experience a summer rental outside of the beach areas is typically far lower priced. And if you rent for 3 or 4 continuous months there's usually a huge discount. With the internet it makes it far easier to find basic, but clean and safe, rentals anywhere.

I just looked online and I found many rentals from July 1 to October 1 on Long Island. Prices for a decent room or an apartment, even hotel room surprisingly started from $900 to thousands a month with many being around $1200 to $1600 per month! These are not on the beach nor are they luxurious but you can see just how affordable they are compared to going on a holiday and spending $200 to $400 per night! And those are the listed prices. I'm sure you can negotiate the prices down. So for two people I would say you get out of the Arizona heat for the whole summer and spend it on Long Island for like $6000 (airfare, rent, bicycles, extra food cost). Just live like you do back home and it should be very enjoyable.