Firstly we want to congradulate you, or your player, on a unique character concept! Frankly we're shocked at the blatant lack of good manners in many of the replies on this post. After all, this character was BORN without arms, he hasn't suddenly had them cut off. There should be no penalties to Dex and basic human functions. He never had to ADAPT, because this is the way life has always been for him.
Under "blinded" in the DMG it states many penalties, but then says, "characters who remain blinded for a long time (like, say, from birth!) grow accustomed to these drawbacks and can overcome some of them (DM's discretion)". We think you should keep this in mind, and that the people who have responded to this post should, as well.
As to grappling, if you've ever watched wrestling or know any capoeira, you will know that grappling with your legs is a plausible option (scissor kick). This typically takes the character to the ground with their opponent. When the character fails to make the touch attack, we would say the character ends up prone, while his opponent does not. Instead of imposing minus' on the actual grapple check, we would suggest you consider this as the penalty.
As for skills, we would suggest, regardless of the character's class, keeping Swim out of class, simply because learning to swim without arms would be doubly difficult. Trapeze artists often hold themselves up by their teeth. We don't see why this can't be applied to Climb, when using a rope (pulling yourself up with your teeth, and twisting your legs to secure the rope beneath you . . . dude, this guy is cool!). Clearly it would be a lot slower, but it works. We don't see why craft is a problem, except, perhaps, smithing. And penalties to Escape Artist!? Hell, this guy should get bonuses. Consider how precise he is with his feet, Forgery might be more difficult, as it is so precise. Some Sleight of Hand operations may be difficult, not cutting a purse, but picking a pocket. We don't see any problem with any of the other skills.
Things requiring manual dexterity can be done with his feet. As for spells with touch attacks, we would suggest you can touch the opponent with any part of your body (headbutts make for a nice visual). Have him possess a bandelier that straps at the shoulder, which can hold potions close to his mouth, so he does not have to "use his foot". That would be ridiculous in battle. Wands, however, can most certainly be activated by foot.
Benefits may include bonuses to squeezing through tight spaces, and maybe to Dex, as he has had to be flexible all his life. In some systems if you take huge penalties like this, you gain benefits in other areas to balance your character. You may want to consider giving this character a bonus to Int, or some such thing.
Remember folks, role-playing is about playing exceptional people. If you doubt the abilities of disabled people to accomplish great physical feats, we suggest watching the special olympics.
Good Luck (meant sincerely)
T, J, and R from Three Haligonians