Have gravestones installed on the server, and for people that have it installed on client side it works great. However, if you are running a vanilla client and die, all your items disappear.

It is not intended to use the mod server side only. You need the mod on both sides. You shouldn't even be able to join without the mod.

So if you are running a server with it, it's expected that everyone must have it installed also or not to play on the server because it won't run correctly?

Of course!

Seems a bit short sighted to me... But what the heck, I'm just the user of the mod

You shouldn't be able to join the server without the mod. Forge should handle that. Because mods that add block need to be installed on both sides.
Might be a Forge bug.

But this is also noted in the FAQ:

https://modrepo.de/minecraft/gravestone/faq