Absolutely, yes.
Historically (and prehistorically), most sharp stone tools (projectile points, scrapers, knives, etc) were made from obsidian (volcanic glass which lacks a crystalline structure), cryptocrystalline silicates (cherts, such as flint), or fine grained volcanic rock (e.g. fine-grained andesite).
Generally speaking, when one wants to make a new tool, the maker will use the percussive force of a stone against a "core" of toolstone to knock a roughly tool-sized flake off of the core. The flake is then be shaped and sharpened via percussive flaking (hitting the tool blank with a rock or antler to knock off smaller flakes), and is ultimately sharpened using pressure flaking (pressing the tip of an antler or dense stick into the tool to knock off small flakes)[1]. The flaked edge can be incredibly sharp (e.g. obsidian flakes to blade whose cutting part can often be measured in atoms of thickness; ages ago, when I was actively learning to make stone tools, I sometimes shaved with obsidian flakes). This general process is called flintknapping.
As a knapped tool is used, the edge will wear (as flakes are knocked off in a non-systematic fashion). 933
The edge can be refreshed by using the same knapping technique as is used to construct the tool in the first place.
A great reference for the practice of flintknapping is Dr <PERSON>'s book Flintknapping: Making and Understanding Stone Tools (disclosure: <PERSON> is a good family friend; the book was illustrated by my mother—that said, it is an authoritative text in the field).
That being said, it is likely that stone weapons will come in one of two general varieties: projectiles (like atlatl darts or arrows), and "bladed" weapons with stone blades embedded into a more durable medium (such as the Mesoamerican macuahuitl). In either case, using the weapon is likely to either leave the stone parts relatively unaffected, or to completely break them. It is unlikely that one would seek to sharpen the edges of such a weapon, and more likely that one would simply replace the stone parts (i.e. replace the obsidian blades of a macuahuitl). Stone itself is generally quite brittle, hence one would not expect to make a weapon entirely out of stone (e.g. a stone sword would likely be quite useless, Minecraft notwithstanding).
[1] It is worth noting that stone tools can be quite useful, and that, for example, it is possible to buy obsidian surgical tools. 523