There are currently two ways to obtain Seafarer's Dubloons:
1) Get them from the caches you get by participating and winning Island Expeditions.Every time you win an Island Expedition, or occasionally when you participate without winning, you get a bag with Seafarer's Dubloons in it. The number of dubloons can vary wildly, but it's often between 3 and 8 coins per cache. With a lot of luck, you can get up to 30 Seafarer's Dubloons in a single cache, but that's extremely rare!
2) Loot them from the ground on the islands.Dubloons caches can also spawn on the ground on the islands in Island Expeditions. These small, brown caches can be very difficult to see, but you find them every once in a while. You need to interact with the cache to loot the Seafarer's Dubloons from it, and only one person in the group can loot the cache. I recommend always picking up these if you come across them.
Most islands should have at least two of these caches on the ground, so keep looking if you can't find any.