Welcome! This is a pretty fun question, because it shows both the strengths and limitations of the game. On the one hand, there’s no full quest line to abolish slavery in Morrowind, although some slaves can be freed. There are some mods for adding quests, but I haven’t looked into them very deeply.
On the other, since no NPC is essential you can end slavery through brute force and kill every councilor that supports it, every slave trader, etc. Bloody, but still arguably in line with the “might makes right” philosophy of the Telvanni.
So while there’s no “official” way, the game leaves the door open to roleplaying if you’re playing a character who would take matters into their own hands.
If you kill an NPC needed for the main quest, the game will give you a “the thread of prophecy has been severed” message. It’s still possible to get the necessary equipment to fight the final boss of the main quest even if you kill, say, Caius, but I wouldn’t recommend killing any main quest NPCs until after it’s completed.
The built-in failsafes aren’t explained in-game, and are buggy. It’s more so just a cool last resort for repeat playthroughs where you play, say, a traditionalist who refuses to work with the Empire to solve the blight storm crisis.
It’s one of the coolest things about the game. Even though I’d never consider methodically killing everyone on the island, the knowledge that you can do it makes the decision not to much more impactful from a roleplaying perspective.