in a word, yes, you can rent a cruiser on the big island.
affordable rent-a-car is on palani road in kailua-kona, they will pick you up from the airport but i don't know that they have a website.
808-329-776
it's a cheap rental (dings come with the territory, but they have a few convertibles as well) and won't peg you as a tourist, if that matters to you.
(it's also worth checking the price at the chain places for the time you're planning to be in town...you just never know.)