Yes, you should be able to get a bike on the island. If not, it's only a 30-60 minute ferry ride (depending on which ferry you take) to the mainland. I recommend Stedman's Bike shop in Wakefield. Website:
The island is very laid back! Really good people and close community. Bring warm cloths for May. It's still chilly some days and at night. Season kicks off on Memorial Day and by July 4th it's really busy. It sounds like you like to explore and there's lots to see on this tiny little island!
There are only a few places that stay open during the winter. The1661 Inn and the Inn at the Spring House, Victorian inns like the Avonlea, even resorts like Island Manor are open; restaurants that stay open include the Barn at the Spring House, Kimberly's, McAloon's, Island Pizza and Bethany’s Airport Diner. The Block Island Grocery and the Block Island Depot are also open year-round.
Where are you working this summer? When do you arrive?