What is inaccessible in a world without mobs?

Although I cannot bring up a full list right now, I did want to mention something that I think would be quite significant if you're planning on beating the game on that world with no mobs; the only way to obtain blaze rods are via blazes, and so if no blazes spawn, you wouldn't be getting any blaze rods needed to craft Eyes of Enders (unless monster spawner blocks aren't effected by the "Disable mob spawns" gamerule, however I believe that it would be unlikely to be the case).

You can enable mob spawning but switch the difficulty to Peaceful so that no hostile mobs spawn but passive ones do. Just be wary of hostile wolves (wolves that have been attacked) as they are still hostile.

Rotten Flesh, bones, meat of all kinds, Phantom membranes, Blaze Rod, Ghast Tear, Ender Pearls, Magma Cream, Spider Eye, Leather and Rabbit Hide, Mob Heads, String, Rabbit's Foot, Slime Balls, Gunpowder, any type of Villager-exclusive trade such as Bottles of Enchanting, and others that I can't think of at the moment.

Leather can be found in village tannery house chests (Unless you disabled structures), and someone else mentioned it can be found through fishing. Rotten Flesh and Bones can be found in chests, Rotten Flesh in Underwater Ruins and I believe Desert Temples as well as by fishing, and Bones in Desert Temples. Spider Eyes can be found in Desert Temples sometimes. Slime Balls can't be found in chests, so you have to rely on taking apart the redstone in Jungle Temples for sticky pistons. Raw Beef can be found in a villager butcher house chest, I don't know if other types of meat can though. Potion Brewing Stands can be found in villages as the Cleric workstation, but there isn't much use to them as most useful potions require mob drops. String can be found in Mineshafts and Abandoned Villages through cobwebs in rather plentiful amounts, I've found around half a stack of string at least in one wander through a Mineshaft. Gunpowder can be obtained in Dungeons, Desert Temples, and Village Chests, and there is TNT in the Desert Temple trap if that's why you want gunpowder.

You won't be able to beat the game unfortunately, but you might be able to find Ender Pearls in Stronghold Chests. Hope this was of help, I'll edit if anything else pops up.