To fit the vanilla feel of having real life animals not drop items gazelles could shed their horns instead of having to be killed.
This could be done through magical means; such as throwing a special potion or feeding them a magical item.

Still unlikely

Gazelles don't shed their horns irl actually. And the main idea behind the vanilla mobs not dropping items is for endangered mobs and gazelles are not endangered.

"in 2016 the population has declined to 2,500" Wikipedia
"It is critically endangered and has a population of less than 500" LiveScience
"with a scant 400 estimated to remain in the wild." nationalzoo

That is not Alex's Mobs gazelle, that is a mountain gazelle. Alex's Mobs adds the Thompson's gazelle which is not endangered at all. You can see the difference from the black stripe along their bodies. And like I said before, Gazelles are one of the animals that actually don't shed their horns.

"Horns are not shed and their bony cores are covered with a thick, persistent sheath of horny material, both of which distinguish them from antlers.[13]"

I've edited the post to say "This could be done through magical means; such as throwing a special potion or feeding them a magical item," but thanks for telling me.