Can you ask them to make you salads without the dressing and without chicken, even? Poach or steam stuff instead of grilling it? Poach or hardboil some eggs for you to put on the salad? Where I work, the cooks are totally happy to do make little changes like that as long as you're not asking for it during the middle of dinner service, and of course provided that you're friendly with them.
Also do you work back of house or front of house? Just wondering since if you're BOH you would be aware of pretty much every ingredient in the food already and might not be so keen to eat it at all. I work BOH in a restaurant too and have accepted that I won't be eating the food, even if it's super cheap, after seeing how much vegetable oil and other stuff goes into most dishes. It's not worth it to me, especially since I don't get it for free, just discounted. Most restaurant food is a minefield.
Depending on how sensitive your stomach is you might make the same decision. If this isn't a fairly high-end place that makes nearly everything, there can be questionable ingredients in everything from the broth/stock they use to the spice mixtures.
Edit: after actually reading the menu throughly, a lot of it looks seriously delicious, and not exactly paleo-unfriendly. Yes, there will be a small amount of canola oil on the surface of the food, but all in all it doesn't look so bad. I would think if you avoid stuff like salad dressings and anything deep-fried, it might be okay for you. Unless, of course, you find even small amounts of canola oil upset your stomach (like you said in your question).