Sea water contains sodium, potassium, bromide, sulfate and bicarbonate ions. Therefore, in principle you could use fractional crystallization to get most of what you would need. It would be a bit like the Curies separating radium from rock, but what's time to someone marooned on a desert island?
Taking some time out to find a volcanic fumarole wouldn't be a bad idea; once you have elemental sulfur, from any source, you are on your way. Having sulfur to oxidize to the trioxide would let you make sulfuric acid, useful in extracting gallic acid and for other purposes. Reducing sea-water sulfate to free sulfur might be a challenge; is there a chemical engineer in the house? (Carbon [i.e.--charcoal] reduction of sulfate might work if the thermodynamics are right, but I don't have a convenient way to check at the moment.)
Presumably, this island has coral, shellfish, or at least carbonate rock available; burning the calcium carbonate to lime will allow you to make calcium hydroxide, which can be used along with wood ashes to produce sodium and potassium hydroxides---burn some of your sulfur (where did you get that, anyway?) to sulfur dioxide as Ole describes and you have solved the fixer problem.
If you don't want to get your carbonate from the water, you could also reserve some hydroxide; left alone, it will absorb carbon dioxide and react to form carbonate, which is (conveniently) insoluble in concentrated hydroxide.
I hope that you had the foresight to land on a largish deserted island forested by oak trees; not only would the ash provide alkali, and the galls, acorns and bark provide developers, but the fractional crystallization is not going to happen by itself by itself, so you are probably fated to spend a lot of time (and fuel) tending a fire!
Any time now some folks are bound to show up and point out that our desert-island chemistry isn't environmentally sound and that our products don't meet Kodak's quality standards; when this happens, we will hijack their ship and sail home to get our film developed there!