But they search containers and pick up items if I place them on the ground. I'm sure its not a weight issue, they just refuse to search dead bodies when I tell them to.

are they capable of searching dead bodies or Is this some sort of bug? does anyone else have this problem?

If they can't search dead bodies is there a mod that enable them to?

Hmmm. It has never occurred to me to consider this a "problem". Probably because the time it takes for me to open the companion interface, and give them a command to do anything I could have easily looted the corpse, moved on, killed something and looted an additional 3 corpses.