Would a prescription, nightly mouth guard raise the risk of cavities? if so, how?

No: The factors underlying dental decay include the patient's capacity to remove plaque and their capacity to limit foods that contribute to plaque formation. Removable oral appliances such as a nightguard, when properly maintained and cleaned, do not contribute to decay.

Not normally: Normal maintenance and care of your night guard and teeth should not raise the risk of tooth decay. I would recommend that you brush your night guard before and after use, and once a week soak it in 1/2-n-1/2 minty mouth wash (not listerenetm) for 5-10 mins. To freshen and clean the appliance. I recommend that the appliance be stored in water when not in use. I hope this helps answer this questio.

One possibility: ..Would be that if you ate sweets or drank soda without brushing or flossing and the night guard was being worn in such a way as to keep sugary residues in contact with plaque on your teeth. This seems unlikely but it is one way the appliance could enhance the decay process.