They could not send telegraph because German's had cut their communication lines.
Following are General's dialogues from tent briefings.
ERINMORE: Colonel Mackenzie is in command of the 2nd. He sent word yesterday
morning that he was going after the retreating Germans. He is convinced
he has them on the run - that if he can break their lines now, he will
turn the tide. He is wrong.
ERINMORE: Colonel Mackenzie has not seen these aerials of the enemy’s new
ERINMORE: The 2nd are due to attack the line shortly after dawn tomorrow. They
have no idea what they are in for. And we can’t warn them - as a parting gift, the enemy cut all our telephone lines.
Since it's not discussed in any length why there was no option of airdrop, my best guess is that they received the new aerials of enemy's lines after the communication was cut off. It was 1917 so definitely someone must have traveled to the General Erinmore to deliver it. Due to loss of communication Erinmore had to resort to postman method to deliver the message to Mackenzie.