To collect the sol emeralds, discover and complete all islands at least once. Then, talk to the captain and he will tell you to inform Blaze that one of the emeralds have been located. You will either have to finish a race or replay the bosses.
After completing the game and fighting at Big Swell, Tails will ask you to get material for him to power up your watercrafts. You must power up the water bike twice to get the Emeralds faster. This excludes Sol Emeralds.