You're absolutely right — the game is missing a proper tutorial and navigation, no doubt about it. What you need to do is go up close to the table in the sheriff's house where the ammo is. Once you're close enough, the ammo will automatically go into your inventory.
Keep in mind, though — the ammo doesn’t go straight into the shotgun. You’ll need to press R to reload it. You can also press I to check if the ammo made it into your inventory.
If none of this works, then it might be a bug I wasn’t aware of — and I’m really sorry about that.