Yes, home ones are. Mine is. But the subject of that paragraph concerns restaurant grills and griddles. Are those propane fed? I don’t know, but I don’t think so.
Ok, you wrote “grill” so I thought we were back to the home grill (vs McD griddle etc). But I can see what you meant.
Nevermind!
Ah, sorry for the confusion. What I meant to convey was even if gridiron-style grills were being used (and some restaurants do have these style of grills, I believe), would the propane permeate the food? I honestly don’t know the answer to the question. It’s being treated by some as a given in this thread, but is it? Completely combusted, propane becomes water vapour + carbon dioxide, right? But does it not completely combust or something? I just wouldn’t have thought propane could permeate the food in any significant way, whether using a gridiron grill or a griddle.
And the second part was that I’m not entirely sure propane is even being used in the restaurant setting. I would have thought that is natural-gas fed, but, once again, I don’t know for certain. (As this happens to the OPs wife in both outdoor and restaurant grill settings.)
My guess would have been improper food handling, as well.
I don’t know a whole lot about propane combustion, but I’d guess that incomplete combustion would be likely to produce soot and carbon monoxide, rather than carbon dioxide and water. I’d imagine you’d notice sooty food 
If you’re going to figure out what the problem is, you need an experiment that separates “cooked on a propane grill” from “cooked by the in-laws”.
Everyone saying that it’s pretty unlikely that propane is an issue is correct. The likely problem is that your in-laws are not safely handling food.
My guess, based on the evidence provided: The in-laws aren’t good at washing their hands or prep surfaces between prepping the meat and the buns/toppings. The reason you didn’t get sick with the grilled chicken is that you don’t generally put toppings or buns on grilled chicken. She doesn’t get sick from McDonalds food because McDonalds’ employees wear gloves and keep the raw meat away from the other food.
Obviously, it could be something else, but it’s extremely unlikely that it has anything to do with the heat source.
Propane burns pretty cleanly on a functioning grill. Having recently fixed a grill that was not burning cleanly, it’s extremely noticeable. The food is covered in soot.
Focusing on propane is going to make you miss the actual danger, which is: You shoudn’t eat any food they prepare when they serve hamburgers.