I am sure that all they do is open a box of ready prepared chickens and put them in the rotisserie to cook, so it might be worth asking to see the package that they take them out of.
If they start with untreated raw chicken they ought to be able to tell you the ingredients of any marinades and rubs they coat them with.
I have had success in persuading serving staff at deli counters letting me read their label info. Just tell them you have some food intolerances and might be made ill if there are any ingredients that affect you.
I also say that I log my food intake to keep an eye on my nutrients so need to know the numbers from the nutrition panel. So far the only reaction I have had is one of interest rather than non co-operation.