I have Klimmek's supplies for High Hrothgar and am standing looking at the chest. The arrow telling me this is the place is right on it. But I cannot figure out how to place the supplies on the chest or in it.

Click on the chest, if on the xbox with the A button. Look to the left side and scroll down to where the supplies are, I believe they are under misc. When you get to them with the arrow pointing to them, click on the X button . I do not know what key to use for the pc.

....um.....you literally click on the chest and put the item in the chests inventory

I am thinking the op is asking for "how" you click on the chest anf put the stuff inside. I know the commands for the xbox but not the pc. Can you give them to the op??

E , just like everything else with interacting, not sure how he can not know how it open chests and still got the quest considering they use the same button and it explicitly says on the screen how to interact with an object.

It must be wonderful to be so infallible.

Well, I'm on a pc and can certainly open the chest by pressing E. I did that and went down my list to "misc" where I know the supplies are and the chest closed right up again without letting me do anything. I shall try again, however. If that's how it's done, then I'll just keep plugging away at it. Perhaps I accidentally clicked on something other than "misc". Thanks for the answer.

Sorry I could not help you further.

E to activate chest and then make sure, double sure, that Klimmek's Supplies are highlighted (I once deposited a kettle ) look at the bottom of the screen it should have an action key listed for Deposit which I believe is E again. When you do, your quest should update if it went into the chest. If it did not then you may have a mod conflict. Cheers!

Bram

Yep. Got it. (And no mods here, by the way.)