I'm wondering if it's reasonable to put one through the edge of a board (first image below),
Yes it's reasonable in theory. Sometimes pocket-hole screws are installed there for various reasons, including your case here where once face of the 'pocket board' is not accessible and the other is a show surface.
However, the previous joint solution here of dowels complicates things. I believe we can see that the screw will transect the dowel at top (top of the image) and because of symmetry it may be exactly the same for the screw going into the opposite edge. This could considerably undermine max strength (but see last point).
I was planning to do one on the top edge and bottom edge.
If it were viable for this situation that would be highly advisable and because the two screws would go into the leg at opposing angles you get a dovetail-like hold, infinitely stronger than if the two screws are parallel as in standard orientation for pocket-hole screws in such a joint.
Assuming this is a good idea, how should I choose the right screw length?
This is really a separate query and should ideally have its own Question but in short the rule of thumb for screws is use one as long as is reasonable for the application. And where strength is a factor, as long as possible, or as thick as possible, or both.
On the plus side here, this unusual location for the pocket-hole screws means you can go much longer than you would otherwise, where longer screw choices would eventually risk the tip of the screws poking through one face of the leg. But instead your screws will angle into a nearly unlimited amount of wood real estate in the vertical axis of the leg, so can be very very much longer than you'd be using otherwise.
The purpose-made panhead screws for use with Kreg jigs are not cheap at the longest lengths, and have limited potential for future use. So perhaps it would be a better choice to go with a different screw (such as a modern construction screw, which has guaranteed future uses). If necessary add a small washer to spread the load within the pocket holes.
Debatably better, repair the existing joint
As a general statement, it's better to repair chairs with breaks such as this by repairing the original joinery than by using screws (of any type and in any orientation).
This will involve:
- partial disassembly of the chair;
- cleanup of the joint surfaces, including the dowel holes (which may involve re-drilling to a slightly larger diameter);
- replacement of both dowels;
- re-glueing and holding under clamp pressure until the glue has set.
You'll need a number of clamps for this, not just one, of sufficient reach to span the dimensions of the chair. White or yellow glue, or epoxy, a drill and a range of bits.
You could also take the opportunity to install a third dowel in the middle if you wanted to. Although, if well-glued and clamped, the existing joint is likely perfectly adequate, the extra dowel would substantially strengthen the joint (possibly adding more than 50% extra).
