There are in fact other mods that let the player lock doors/containers, but I had some specific ideas for how I wanted this to work in my game, and thought I'd share the end result with you all.  It's mostly a small immersion mod but it feels good to lock up your homes/gates when leaving, and it's nice to have the Skeleton Key be more functional.

Big thanks to Dylan James and Dissonant Values in the /r/SkyrimMods Discord channel for giving me ideas for the Skeleton Key implementation.

There are no requirements to use this mod.

HOW IT WORKS

On game load, you are given a shiny new key ring.  When this key ring is equipped and your weapon is put away, you can lock any door as long as you have the key.  Doing things this way allows you to enable the functionality only when you want to use it.

If you want to lock a container, you must craft a padlock at the crafting forge under Misc, and place it inside the container you wish to lock.  With the key ring equipped, it can be locked.  The key ring also acts as the key to unlock any chest you have padlocked.

If you obtain the Skeleton Key during the Thieves Guild questline, you can now use it to Lock or Unlock any door, container, or display case once per day while the key ring is equipped, and the Skeleton Key is in your inventory.

THINGS TO NOTE

• Locking personal containers isn't useful beyond roleplaying, but it sure feels good!
• Extra key rings are craftable at the forge under Misc as well
• Skeleton Key should be USED AT YOUR OWN RISK!  I am not responsible for any broken quests or consequences as a result of unintended use!

INSTALLATION

• Install any time after exiting Helgen, or beginning your alternate start.

UNINSTALLATION

• I do not recommend removing mods during gameplay, but you should be able to remove this without any lingering effects.

SHOUT-OUTS
Just wanted to give a shout-out to the original Lockable Locks mod, as well as Skeleton Key is a Key SE, which I just learned of as I was publishing this.  Check those out if you want some alternative implementations!

Skeleton Key is a Key SE
Lockable Locks

Enjoy!  Let me know if you experience any issues.