Sorry, I tried looking everywhere but I just can't find how to do it. I can delete things by pressing delete but there's no right-click menu to add items. Please help!

You must use the good old "drag and drop". Open the npc form, chose in the item list the item you want to add (for example a sword in "weapons") and drag and drop it into the npc inventory.

Ah, thank you so much. You would think a developer tool like TESCS would use something more complex that Drag+Drop, but no. Thanks again.