Yes, a snoring mouthpiece can work for a person who is a mouth breather. If you select a boil and bite or custom mouthpiece, choose one with a breathing hole in the front of the mouthpiece. For the boil and bite mouthpiece usually you will see a removable air way plug which is used during the fitting process to keep the airway hole open while the material softens.
Then if you are successful with the boil and bite mouthpiece, you can convert to a custom laboratory fabricated stop snoring mouthpiece. Custom mouthpieces should have an airway opening in the front to allow for mouth breathing, if you have any form nasal obstruction.