Mulch will not cause any serious illness with your dog and the dog should be perfectly fine if they ate some mulch.
As long as you have no fertilizer on the mulch the mulch itself is not harmful to the dog although it might block their bowels up or cause the dog a stomach ache.
Try to keep the dog from eating mulch though because it could lead to a bowel blockage and constipation if they happen to eat enough of the mulch.
But just a little bit of mulch should not be a problem and they might vomit it up or feel bad for awhile if they ate too much of it.
Just try to keep the dog away from the mulch and when they try to eat the mulch tell them that they shouldn't be doing that and remove them from the area.