I am a sitter however I do not do boarding, if I did do boarding, yes, I would recommend the owner of the dog, bring the dog's food.
As mentioned dogs can be allergic just like humans to foods, the owner usually is aware of allergies, or foods that their dogs love to eat and will eat. Snacks should also be purchased and brought along with the owner and their dog.
I usually bring biscuits when I first meet new dogs at the owners homes, but I always ask the owners before I give a dog a snack i have purchased.
I think it is a good idea to bring along a favorite blanket or bedding that their dogs sleeps on, so they feel comfortable while mom and dad are away. Always ask the sitters about bedding.
You might want to ask your sitter, but it is best if you bring your dogs food to the sitters house. Changing his diet can cause digestion problems like loose stool. I wouldn't bring any toys, as your dog can get aggressive over them if there are other dogs at the sitters home.
It is best for the dog owner to provide the food to insure your pup gets what they are used to.