Keycards are found on the final level, Risk of Rain. They are dropped randomly from enemies or found in golden containers that look like the caches of xp and gold you open in other levels. I suggest stocking up on 56 Leaf Clovers to increase item drops from Elite mobs.
Update: It has been confirmed that all 4 keycards must be obtained in one run, so you can't collect these over multiple attempts. Thanks LifeOfDaPartay3!