I'm trying to add a custom item to Minecraft that players can use as a currency (e.g., a gold coin block). Can I do this with a resource pack, or do I need a full client-side mod? I haven't been able to find anything about it online.
No, adding custom items using resource packs isn't possible.
Resource packs only provide a way to modify certain aspects of in-game elements, according to the Wiki:
The resource pack system provides a way for players to customize textures, models, music, sounds, languages, texts such as the end poem, splashes, credits, and fonts
However, maybe what you're looking for is a data pack. Data packs cannot add completely new items, but can be used to add modified versions of items that, when combined with resource packs, look like a completely new item.
I found this tutorial on YouTube showing how to add "custom items" through data packs, maybe you'll find it useful: