Is it safe to use just the chippers in a dado stack?
No. The chippers are designed to do just that — chip out the middle of a groove being cut on either side by both outside blades.
Some dado stacks come with instructions that state not to use the chippers without both blades, example from Freud, sixth bullet point from bottom: "Always use both outside blades of the dado set. Never use the chippers by themselves or with only one outside blade."
It may be the case that all dado stacks are only safe to use this way (and I strongly suspect this is the case) but that the instructions aren't as well written for all sets.... if, for example, your set doesn't explicitly say this.
In case it's not obvious for any future readers, the opposite is not the case — it is perfectly acceptable (and common practice) to run the outside blades by themselves, without any chippers, shims or other spacers in between. See both diagram and chart in this PDF from Diablo1.
The correct way to cut a needed groove using a dado set of any kind that can't cut it in a single operation is to use multiple passes. Generally you'll be able to cut once, then move the fence and run a second pass to get the width required2, but you can cut a channel of any width this way if you just keep going.
Alternatively, just use your router
This is bread and butter stuff for a router, and of course far safer3.
realize I could pick up a 5/32" router bit to make the cuts
Although at first glance it seems very handy4 it's not required to have a router bit that matches your stock thickness, since a smaller bit can always cut a groove wider than itself via multiple passes.
but that would require making something of a jig to help get nice, straight cuts through the middle of the boards
No jig is required for this, at the most basic you just need to run your router along a straightedge.
If added security against wandering is required you can arrange a second straightedge parallel with the first. This doesn't take as long as you'd think, and once you've done it a couple of times it's something you can set up almost on autopilot (since the base itself is used to ensure spacing there's no measuring needed for that part of the setup).
1 Notice that it is implicit in this that both outside blades are fitted for all grooves wider than 1/4".
2 Rather than relying solely on the tape measure built into your saw or measuring manually from the blade to set the fence for the second cut scrap pieces should be run so you can dial it in; this should probably always be done, but it is vital where very tight tolerances are required.
3 Important reminder that dado stacks are considered so dangerous elsewhere in the world that in those markets they're actually banned from being fitted to table saws in commercial workshops. And furthermore, saws are deliberately fitted with short arbours that makes fitting one impossible.
4 While your router bit and stock may be the same size on paper, the reality might be quite different. The bit is likely to be dead on, but the stock could be way off (particularly with plywood, which not only can be over- or undersize, it can even vary in thickness place to place).