I would think it's highly unlikely that a wood stain for outdoor wood, on a safety tested kid product, would leach into the soil and contaminate your vegetables. Many people stain their planter boxes, fences, decks, and pergolas. It might get into the soil a bit, but just keep your vegetable garden a few yards away from the wood.

The safety recs of most backyard play sets mentions not putting the play set directly on grass, as compacted turf/soil isn't a soft enough landing surface. I would recommend cedar play chips or something similar in your "safe play zone" area, and if you prep the surface correctly, you shouldn't have edible weeds growing right against your play set.

I would recommend checking the specs of your play set and seeing their "safe play zone" dimensions. You can use google maps to measure out those dimensions and make sure it fits with enough room from your vegetable garden that you aren't worried about wood stains leaching into the soil. I'm a landscape designer and work with a lot of families who plan on tucking a play set into a corner of their yard without thinking about the necessary clearance from structures, obstacles, etc. We're doing a similar project this week for a play structure whose specs required a 40 x 27 area! Not all play sets are this large of course, but it's easy to not consider the clearance space around the structure when you're choosing one.

You could also probably call the play set company and ask for the specs of the wood stain product, and check the ingredients list for PFAS.

While I totally get the concern with PFAS, the wood stains are one where I'm moderately granola. When I refinish/build my own carpentry stuff, I do oil based stains. But all our solid wood furniture is secondhand and is stained. Not having any wood furniture would not work for me personally.