You would have to check with the manufacturer if they can be run 24/7, I doubt they are designed to do this. You could always wire it up with a plug on the end and use a wifi/smart socket to set a schedule for it to come on and off.
We have stored plenty of stuff in sheds and outbuildings in cardboard (used to live on a farm) over the years and never had much issue with damp and none of those were heated so I don't think that just because its unheated you will have damp, however heating as others have said would help but obviously its not very efficient to heat an uninsulated outbuilding, not to mention any fire risk you introduce by using a unattended heater.
What's the roof material? if its tin you could be getting condensation forming on the underside giving you moisture/ drops on to your stuff below?
What's the ventilation like currently? before using anything powered I would look to try and improve the natural ventilation first, we often try to block all holes up in garages to stop mice and this reduces the natural ventilation. I would look to try and have a mix of low and high level vents with vermin grilles to try and get some natural ventilation in there first.
Do you use the garage for anything except storage? such as washing machine or tumble drier or anything like that?