How come the Vale never tried to take any of the major islands in Blackwater Bay, like Dragonstone? Stormlanders are not noted for being a seafaring people, and the Ironborne and Redwynes are on the other side of the continent. So who stopped the Arryns from taking them?

I’m sure it tried at some point, but wasn’t able to project power from a distance for very long, especially against opponents who would have shorter supply lines and more strategic depth. 

Dunno where you got the idea that there weren’t seafaring powers in that area: 

“The stormlands have also produced their share of great seamen and sailors. Storm’s End itself, looming over the great cliffs of Durran’s Point and the treacherous rocks of Shipbreaker Bay, offers no safe anchorage for either warship or merchant craft, but in the time of the Storm Kings, war fleets were oft maintained on Massey’s Hook, Estermont, and in the towns and fishing villages along the Sea of Dorne. Later, other monarchs preferred to dock their fleets on the western shore of Tarth.”

That’s the Masseys, the Bar Emmons, the Estermonts, and the Tarths, and the reference to the sea of Dorne coast suggests the Mertyns, Swanns, and Wyldes as well.