There are three lockpicks in the tutorial dungeon. One in the first large room with the center tomb - one of the chests you try to open will say something about looking for a lockpick, and it's right next to the chest. Bashing that chest is not advised, it's got a high durability specifically to encourage you to use the lockpick, I'd imagine. The other two are in the area with the burning crates that you need to use water barrels or the rain scroll on. One will be on the opposite direction of the chest with the trap on it, sitting on top of a crate. The other will be near the large stone door, by some searchable containers.
Use ALT to highlight objects and see them more easily.
You can also make 4 lockpicks with hammer + 9 inch nails, and you can reliably get nails from the mortician.