George Mulhauser Plycraft lounge chair  

Additional pics

I tore apart a Mr. Chair and had it reupholstered last summer, it was a challenge to my upholsterer, good luck!

I've refinished the wood on a bunch of Plycraft lounge chairs. Test the finish first---some were done in tinted lacquer which is relatively easy to remove with lacquer thinner or acetone. If you have nail polish remover in the house, that will do for testing--just dab onto the finish and let it sit for 20 seconds or so, then wipe off. It evaporates very quickly so drip some more on if it dries out right away. If it's lacquer, you will be able to wipe it off down to the bare wood. To strip the entire chair, lay a single layer of cheap paper towels over a couple square feet at a time, saturate with acetone or lacquer thinner, let sit for a few minutes (rewetting as necessary and this is easier if you put some of the solvent into a small squirt bottle). Wipe clean, then wipe again with fresh paper towels and solvent until all traces of finish are gone. Use a respirator and heavy stripper-resistant gloves. If and when your fingers start to feel really cold, take a break and warm them up!
If the finish tests negative for lacquer, it'll be varnish or maybe polyurethane (not sure when that became the finish of choice). I think I tried Citrustrip on the one that i did that was not lacquered---maybe some other alternative stripper? Anyway, the finish did not come off very well so i switched to the most powerful methylene chloride stripper that i could find and it came right off. Follow all precautions on the container when using this stuff.
Are you doing vinyl or leather? Or fabric?
Check the bottom of the chair for cracks in the plywood. I don't know about this model but the Eames-style ones are really prone to cracking.

Took it apart last night. No big surprises. I think with the wood refinished, and the one small tear in the black vinyl repaired, it will make a nice lounge that I can have around and not worry about the toddler ruining.
I am wondering how best to treat the area on the bottom of the chair with regards to finishing, as it is clearly a lower-grade wood.

Don't sweat the bottom panel. Finish it the same as the rest. Nobody but you and your toddler will ever see it. The thin strips of plywood on the base would concern me more as far as little ones go, though. The ones I've seen have looked pretty splintery on the edges.
The only thing I might suggest would be to add some wide fender washers when remounting the base to help distribute the stress on the plywood shell. As spanky mentioned, this is a common area of failure with these.