How to wire a normally open switch to a recoil start engine with no external battery?
You don't want to be relying on a battery-powered circuit to actuate a relay that will kill the engine, if the battery is dead you lose the ability to stop the engine. Ideally a safety circuit is wired as Ron suggests so that is is failsafe and ceases operation of the circuit is opened, but that's not how stop switch on these engines work, they short a winding in the magneto that prevents the generation of a spark at the plug by diverting the generated current through the kill switch.

So you need the switches to be normally open such that if either one is closed it will kill the engine. To get the logic correct, you then wire the two in parallel.

If I remember right, the small rocker switch on my weed trimmer has the markings the wrong way around, it is biased to the '0' open side as you'd expect for a NO switch, and you push it to the '1' closed position to stop the engine. If you have the possibility of inverting the switches so that the markings or physical position are reversed, you may be able to achieve your intent without any additional componentry - the rocker on the small rocker switch typically will pop out of its housing with a screwdriver/knife blade pushed down either side to flex the body out enough. Looking at the photos - is the switched biased off? It's in the on position.