I just hit level 10 and would like to craft myself some dwarven armor. I'm only in the companions so far, and really haven't traveled far outside of Whiterun.

What's a good dungeon that I can get to easily, where I can grab some dwemer items so I can make myself some armor? Preferably not a super huge dungeon. I seem to recall doing one early on with my last character, Mzinchasomethingorother, but he was a thief so it was probably part of the thieves or DB questline.

Avanchnzel is a good size and if you visit Riften has a nice quest attached to it.
Mzulft has a Dwarven Storeroom outside the main ruin full of Dwemer ingots and junk but empty of inhabitants.

This, but the bulk of them are behind locked doors (not all though)

But 1 of the doors is only Apprentice and theres 17 ingots behind that. I admit the Expert and Master locks might have to wait.

Those two are the ones i always go for first. The ones connecting to Blackreach i explore as i activate the elevators nearby them, so that's a whole lot of dwarven exploring then

Apart from those three, Avanchzencel and Mzuflt (that is College related), i don't think here are more? And Sightless Pit near Winterhold, actually.

I almost always hit the one under Markarth first. To me the enemies aren't that bad, decent loot, you have smithing and shops just outside the dungeon and it also has a quest involved with it from the wizard/scholar. You also don't have to worry about finding it and random encounters since you can take a cart from Whiterun to it and be safe. It is a fun dungeon for me as well.

Alftand, Your character can walk in and exit without screwing up any quests. The dungeon has a lot of locked chests and at a lower level you can get some of the harder to find enchantments. Clearing the Dungeon early also makes the Main Questline and Discerning the Transmundane easier to accomplish.
EDIT If you just want Dwemer stuff to make Armor with, Radbthar is North of Whiterun, and the first room has enough loot to construct a full set of Dwemer Armor, after being smelted.

Have a lof of lockpicks when you're at the lower levels. Opening a Master lock will help with the lockpick leveling when you're at the lower level. I haven't put any points with the lockpick skill. If you're careful, you can open a Master lock on one attemp which I have done so. I also have broken many lockpicks (5-10) in opening a Master lock. If your'e not aware, when the lockpick breaks it doen't reset the lock at all which helps. Good luck.

Thanks, Raldbthar was perfect! Had to carry some of that heavy stuff, but it turns out it really doesn't take all that much to make a full set.

Eversince the companions sended me to shimmermist cave when i was lvl 6 i have a phobia on everything falmer and dwemer related tho a good dwemmer rune probably is the one east of riften where you have to return a lexicon i forgot the name , lots of dwemerautomations but no falmer. Thank the divines