We have 3x4TB 3.5" HD's in RAID 5 (SoftRAID) in a 2012 MacPro tower.
Is it possible to physically remove them and insert them into a larger enclosure and it will continue operating?
Yes, it is possible to move a SoftRAID array of disks from the internal drive bays of a MacPro and into an external enclosure connected to the same computer. You might have to do a small reconfiguration to ensure that it "finds" the array, but if you have used proper disk labelling it should be pretty much automatic.
Note that it is important that the external enclosure supports JBOD - i.e. it must be able to present the drives to macOS as seperate, single drives. If the enclosure only supports its own form of RAID, it is still possible to use the old disks - but the data on them will be overwritten.
Another thing to observe is that the 2012 MacPro tower by default hasn't got Thunderbolt or any of the newer, faster external busses. You might find an external enclosure too slow if it is connected via USB 2.0 for example.
Before you embark on anything, remember to take backups!