How can I fix/glue/strengthen a small child's wooden rocking chair, where the spindles are somewhat loose?

WalMart. Gorilla Glue. Fix it and forget it.

Get a high quality wood glue and a bar clamp.

Pull the spindles out of their mortises, apply the glue, replace the spindle, clamp it up, and leave it overnight to dry.

The spindles tend to get loose (in time) because they are a bit shorter than the distance from (the end/bottom of the) hole to (the end/bottom of the) hole, in order to put them in (by the guy who fabricated it).
If you are able to deconstruct it and take the spindles out, you could put some wood glue in the holes and replace them.
If not, you could try to squeeze some glue in the space where the spindles enter the holes.
But then I would advice you to first treat one side (let’s say where they are stuck in the seat) and after you let it dry (one day about, to be sure), do the other side (turn the chair upside down first).
This way the glue won’t drip out of it, which would be messy and make for less fixated spindles.
Good luck (it is indeed a fun job)!
Edit: The first one side and turn chair upside down also applies to the first method, by the way.

Elmer’s Glue works good on untreated wood. Rebbel’s technique was spot on. I couldn’t improve on that.

I’m with @rebbel I would completely disassemble the chair and re-glue everything and square it up so it rocks straight. If some of the joints are loose, for sure there are others that will be soon. Better to hit them all so you don’t have to go back and do it all again. Plus this is a good time to sand everything while it’s apart so you get everything to accept the finish.

Gorilla Glue bubbles up, expands and if you’re not careful, creates an ugly dry-bubble-foamlike residue oozing out of the seams. I love Gorilla Glue, but it has to be watched and wiped away before it dries.

I wouldn’t use Gorilla Glue, for the reason @linguaphile says. It expands waaayyy more than you think it will, and will make refinishing the chair more difficult. If you can take it apart, just reglue with wood glue. If you can’t, you can use a glue injector to squeeze the glue into tiny spaces.