Believed to originate from a small area on their foreheads, elephants emit low-frequency sounds that may be used as a secret language to communicate with other members of the herd.
(A) Believed to originate from a small area on their foreheads, elephants emit low-frequency sounds that may be used (incorrect: modifier error)
Believed to originate becomes a modifier of elephants. Incorrect modifier placement
(B) Elephants emit low-frequency sounds that are believed to originate from a small area on their foreheads, and they may use this (incorrect: pronoun error)
Pronoun this refers to sounds though sounds will be considered plural however to get rid of reduncdancey and meaning error, no pronoun should be used for sounds.
(C) Elephants emit low-frequency sounds, believed to originate from a small area on their foreheads, that they may use (The best option)
The middle part that is believed to is a modifier of sounds and another modifier that also modifies sound hence they together convey the intended meaning.
(D) Originating, it is believed, from a small area on their foreheads; elephants emit low-frequency sounds they may use (incorrect: punctuation error)
Semicolon, replacement of and, that is used to connect two independent clauses. Before the semicolon, there is no clause.
(E) Originating, it is believed, from a small area on their foreheads, low-frequency sounds are emitted by elephants that may be used. (Incorrect: modifier error)
Originating represents going on action but this passage requires an ended or source of action thence incorrect