An interesting situation came up recently. I run an summoner with an eidolon that is designed to look like a chain-chomp. As such, it does not have any limb evolutions. While I was aware that they could not wield weapons without said evolution, according to the GM my eidolon also would not be able to use the disable device skill, or at least wouldn't be able to use the tools and would take a penalty. I couldn't really find any information one way or another in searching the forum/prd, so what do you all think? What actions are hampered by a lack of the limb (arms) evolution?
This is interesting and a question I'll be paying attention to, since I've always wanted to make a Dhampir Summoner whose Eidolon was just like a big, hopping coffin because that's funny.
Hmm, my guess: UMD (no arm to wield), Climb (penalty), Craft (penalty), Disable Device (penalty), Sleight of Hand (penalty). I am thinking of a certain critter that's not open source that was just a floating ball (one could say an eye ball) that was able to use it's mouth and tongue to do skills, and I would think that a chomp chain could use the chain/tongue/mouth to do a fair amount of skills at a say -4 penalty.
I asked a similar question about eidolons and hands-free skills a while ago. It's pretty clear than a limbless head can't use thief's tools, so opening locks is going to be tricky, but other than that a GM could be justified in any interpretation of Disable Device accessibility. They could be as lenient as letting you disarm traps at no penalty, or as strict as disallowing any use of the skill at all.
In short, Expect Table Variation (tm).
As for other skills, UMD is fine as long as you have some means of drawing and holding the wand/scroll. Climb and Sleight of Hand both explicitly require actual hands, though you can also evolve a climb speed to gain use of the former.
I assume your eidolon has either the serpentine or aquatic base form, since otherwise it would at least have legs.
For those forms, the lack of arms could reasonably cost you a bunch of magic item slots. I am not just talking about the obvious hands and wrists slots -- arms shape a creature's form in a way that confirms that they have shoulders, chest, body, and arguably belt slots. With arms, a case could be made that an aquatic or serpentine eidolon has every slot except the feet slot.
Now you know why I decided early on that my serpentine eidolon would always have arms. Without arms, an eidolon can basically talk, move, and fight -- fine physical manipulation becomes difficult if not impossible.