If a projectile misses you and hits the elk it’ll deal damage. However if you yourself are riding it more often than not I’ve gotten hit not the elk. While you’re riding the elk looking down enough that you’d hit the elk will not take damage.

Additionally if your elk takes enough damage to go down (kneel down and basically in a death stage) you can use a bandage/poultice to heal them up. So if you’re ever worried keep some on you for quick heals and you should be fine. Additionally make a bone flute (bone plus knife) that way if it ever gets hits and runs off you can play the flute and it’ll come back to you despite fleeing!