Suggestions on fixing this chair

fill it in with stainable wood filler

Wood glue in the crack and clamp the piece ACROSS the crack in at least 2 places to tightly pull the two separate pieces of the seat together. Wait at least 24 hours (follow drying directions on the glue used) before unclamping. I would also add a series of stainless steel flat mending plates screwed in across the repaired split on the bottom of the chair for additional strength. See here, for instance, https://home.howstuffworks.com/home-improvement/home-diy/projects/how-to-repair-wooden-furniture3.htm

I would take the chair apart, drill 3 dowel holes in each side of the split, glue dowels into the holes on one side and let dry before painting glue on both sides of the split and in the empty dowel holes. Put the dowels into the holes and clamp the seat together for 24 hours. You could also put braces on the underside of the chair that will help keep the two sides together.

Wood glue won't hold. And it would be tedious to try to line up dowel holes. Best and easiest solution would be to remove the seat. There should be screws holding it underneath. Use a two part epoxy on the crack and clamp both parts together. Remove squeeze out with nail polish remover. Let it dry overnight. When dry use three mending plates on the underside across the crack for added strength.

Here's some ideas: https://www.hometalk.com/search/posts?filter=fix%20wooden%20chair

I moved to S. TEXAS and my coffee table split like that due to dryness. I brought it home and applied a little steam to swell the wood and then after it dried in the house with a humidifier I used a wood finish oil to finish the repair. Can't find the split now

If you’ve a good carpenter in town, I’d be tempted to ask him to cut some butterflies for and into the underside in the seating stress area. And borrow his furniture clamps

I was afraid to ruin it so very lightly with my clothing steamer and then let it sit in the house not far from the humidifier. After a few days I saw a big difference that is when I used the Furniture Finish Oil in the same shade and later that week I didn't see the crack anymore. Still using that table 17 years later. I will never store it in a dry area again. I bought the humidifier when I bought my piano because they told me if the sound board dried out it would ruin the piano.

,glue, clamp, and pocket hole screws or mending brackets, finish as desired

The wooden seat did exactly what wood does naturally...split on a grain line. If you can, remove the seat. As someone already mentioned, it is likely attached with some screws on the bottom.

Once you have the two pieces removed, apply a good-quality wood glue using a small brush, carefully, apply a layer of glue to both surfaces to be joined.

If you intend to stain or paint the chair now or in the future, be certain to remove any glue that gets brushed onto the top/bottom surfaces of the seat. Stain and paint will not adhere to surfaces that have dried glue. You can use a dampened cloth to remove any excess glue.

If you have access to large clamps, line up the two pieces and clamp, across the break line, in two places. About 4 inches in from the front edge and the same distance from the back edge. Check to make sure the seat is not "cupping" upward. If clamps are not an option, see if you can find several heavy-duty elastic straps, such as thick exercise bands. You can use those, too, but it is even more important to keep the seat as horizontal as possible. Straps have a tendency to pull.

Leave it clamped for 48 hours to ensure the glue thoroughly dries.

I wouldn't recommend using brackets, dowels or similar securing devices. If it is glued properly and allowed to dry sufficiently, it should hold, especially since it's a child's chair.

To give you some information on my advice, my husband and I owned and operated a furniture restoration/construction business for 30 years. We saw damage far worse than this little chair.