You are looking for a noun form, I think.
Your original idea should be "the road running round the island" or "the road that runs round the island".
The general term for the road that runs by the sea is "the coast road", you find these on the mainland too. If you want to stress the fact that it goes all the way round the island you could call it the coastal ring road, though for many readers this will remind them of their city's ring road, not a nice thought when you are writing about a quiet island. 'Circular coast road' would do instead.