Naturally, no store near me sells "tool steel", but some home supply stores sell sheets of fairly thick steel plate.
What's the difference between that and proper tool steel?
Actually we can't say, because there are a wealth of different types of plate, including hot-rolled and cold-rolled, higher-carbon (generically, these could all be classed as 'tool steel') or low-carbon which is mild steel
If I grind an edge on a piece of steel plate, and then anneal it to harden the edge
You misunderstand the word anneal. Annealing is a softening process, not a hardening one; with hardenable alloys of steel you harden first1 and then anneal by a given amount to soften, to produce a steel which is some balance between hard (and therefore wear-resistant, but also brittle) and more flexible (AKA 'tougher', more resistant to edge chipping or outright snapping).
Assuming the sheet steel you can buy is mild steel it's not capable of being hardened. Mild steel can be altered to make it hardenable, but if you're asking this question you're more than likely not set up to do this2, and besides it's not really necessary – see bottom.
would that be sufficient for wood working, or would the alloy simply not be appropriate?
You can actually make certain tools from mild steel (typically various types of scraper rather than true cutting tools) but this would be for limited use and particularly in woods softer than rosewoods, which are hard to very hard depending on subspecies.
For what you want to accomplish you do actually need to use hardened tool steel or it's not going to work well. And that's OK because most of the homemade tools/jigs similar in form to pencil sharpeners simply make use of existing edged tools as the cutter.
This includes chisels for the smaller ones, through spokeshave or block plane blades for those in the middle range and full-size plane blades for the largest sizes.
The blades can be held in place by an arrangement of screws (pinching it in around the edge), by improvising a screw locking system with one or two screws and washers (ideally machine screws if the tool is intended to have a long life), or sometimes simply by clamping the blade in place (IMO not advisable if the dowel will the spun under power).
1 Which if you do it right makes the steel as hard as it can possibly get, often referred to as "glass hard" (and it's just as brittle as that name suggests).
2 Requires e.g. welding equipment, case-hardening compound, a kiln or furnace.