Mosses have a natural anti-freeze that allows them to survive extreme cold, they can however become encrusted in snow and ice, which would impede collection. In general, mosses can be collected year round, it is only the conditions and presence of ice that will complicate things. There are many species that can grow on stone, some of them may be easy to remove even when temperatures are below freezing.
If the moss growth on a stone is thick and you are able to peel it away from the stone without tearing, it is likely you will meet with success. Some species are specialized and grow only on stone, these species attach themselves firmly to the surface and are difficult at best to collect without shredding.
The stone specific species like Grimmia’s are very slow growing and sensitive to changes in their environment. My advice is to collect a small amount and test to see the success of your technique and intended use. Remember to collect responsibly, leaving more behind than you remove. Be sure to have permission from the land owner before collecting and never collect from public property or protected areas.
There are many species that will colonize a stone surface, if the environment is moist, your chances of successful collection and transplanting are high. If the climate is more arid, then tread carefully, mosses adapted to dry conditions can be very specialized and almost impossible to relocate. If the moss is rather easily removed from the stone surface intact, it is a good bet on a successful transplant. If the moss is strongly attached and comes apart when removed it is best to leave it be.