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Who actually gets to issue challenges and accept them? Is it only independent characters? Is it only sergeants and independent characters? Can anyone, even a grunt in a squad accept the challenge? It can't work in out in the rulebook.

Any model that is a character can issue or accept a challenge. All independent characters are characters and many upgraded models are characters such as the sergeant.

Good rule of thumb, if there's a model in a unit singled out with its own stat line, it's probably a character. Attacker gets to issue first, and if they choose not to the defender may issue a challenge. The issuer must choose a model that's giving the challenge, and the defender may choose any character to accept it.

An even better rule of thumb is the reference section at a the back of the rulebook, which will tell you whether your model is a character or not.

Quick clarification to the plusjeremys post:
Challenge may only be issued by engaged character and the challengee may only choose engaged character to accept it.

This also means that if challengee declines the challenge, character challenger selects not to participate in the combat must also be engaged at the start of the combat.