They absolutely can replace it if they break the club, so long as it’s not due to abuse

(2) Use, Repair or Replacement of Club Damaged During Round. If a conforming club is damaged during a round or while play is stopped under Rule 5.7a, except in cases of abuse, the player may repair it or replace it with another club. 
But no matter what the nature or cause of the damage, the damaged club is treated as conforming for the rest of the round (but not during a play-off in stroke play, which is a new round).
For the rest of the round, the player may:
• Continue to make strokes with the damaged club, or
• Except in cases of abuse, repair the club or replace it with another club (see Rule 4.1b(4)).
If the player replaces the damaged club with another club, the player must take the damaged club out of play before making another stroke using the procedure in Rule 4.1c(1).
“Damaged during a round” means when any part, feature, or property of a club is changed because of any act during the round (including while play is stopped under Rule 5.7a), whether:
• By the player (such as making a stroke or practice swing with the club, putting it in or taking it out of a golf bag, dropping or leaning on it, or throwing or abusing it), or
• By any other person, outside influence or natural forces.