Caveat: I haven't dealt with Macs that old since... not too long after they were current, so I make no promises about the accuracy of my memory. You might get better answers over on the Retrocomputing stackexchange.
The first thing I'd do is check the clock battery for leakage. It's a big 3.6V lithium cell (non-rechargeable) on the logic board; I don't remember if it's socketed or wired in on the IIvx. If it's leaked, remove it and clean the motherboard as best you can (including washing out crud that's hiding under chips). It'll run without the battery if necessary, but the clock'll reset (to 1956 IIRC?) and it'll lose a few other settings every time it's powered down.
For mobility: Mac OS has no copy protection (Apple wants to sell you the computer; once you've bought that they're happy), but compatibility between different hardware models and OS versions is a bit of a maze (and some models need a special "System Enabler" file under some OS versions). The IIvx could run Mac OS versions 7.1 (with System Enabler 001) through 7.6.1 (see this and this), so your drive might have any version in that range on it. If you know the specific version, that'll help you search for a compatible replacement. Depending on the "obscure program"'s compatibility, you might also be able to migrate it to another Mac OS version.
You might also be able to find an emulator that'll run what you need, but I don't have any experience there.
Copying the contents of the disk is pretty easy provided you do it with Traditional Mac OS (i.e. before OS X aka 10.x, which is actually a completely different OS). Just use the built-in utility to format the new disk (IIRC it was called HD Setup under later versions of Mac OS, not sure about 7.x), and drag-copy the files over. As long as you copied the System Folder and all its contents, it'll make the new drive bootable.
Compatibility with OS X and other OSes is more limited. Those old versions of Mac OS used the "Mac OS Standard" volume format (aka HFS), which was replaced by "Mac OS Extended" (HFS+) starting in 8.1; later versions of Mac OS and early versions of OS X still supported Mac OS Standard, but I'm pretty sure that died off quite a while ago.