Can you acquire Realm Gold (Realm Of The Mad God in-game credit) without spending real money?

As it is not tradeable, there is no way to obtain Realm Gold except through real money purchase.

As of 2018, the Discord Server has been running contests about twice a month that can reward up to 2500 Gold.

These contests are generally creative-based contests, such as creating new items, creating item sets based on already existing items, writing songs, or creating dungeon concepts. You'll be able to see them all in the #server-events channel, and instructions for how to enter are there.