send a telegram電報を打つ is an idiomatic expression meaning to send or send off a telegram. The て-form 電報を打って connects to what follows or forms a polite request with ください. This expression is old-fashioned because telegrams are rarely used today.
JA: 急ぎの用事なので、家族に電報を打ってください。
EN: It's urgent, so please send a telegram to my family.