A permit or certificate is not currently required just to own a firearm (pistol, rifle, shotgun, other) in CT. They are generally for purchasing a firearm or having a firearm transfered to them in CT.
People can move into CT with CT legal firearms without having to register them or have a permit/certificate for them. One just cannot be a firearm prohibited person (ie like a felon).
They may want to call a CT FFL and drop an inquriy on blank firing guns to see what FFL’s think. The laws are a bit vague on certain things.
Note that this site’s Commonly Asked Questions About Firearms in Connecticut FAQ has an entry addressing moving to CT with firearms.
I’m moving to CT, what firearms can I bring in?
Because CT’s laws are constantly influx, what one can bring in can vary from year to year or even month to month. Consult the CT Assault Weapons Ban statute (see information near top on this page) to see what firearms one can bring into CT. Typically one has 90 days to either modify the firearm so it complies with CT AWB or one can remove the banned firearm from the state.