Is it possible to put a chain on without removing the rear wheel?
It depends on the style of master link and how much chain tension is in your setup.  An SRAM power link needs about 1-2mm of chain slack to unhook the link.  If you can find this amount of slack, it should work with most types of master link.  If your chain doesn't have 2mm of play, but isn't too tight, a master link that snaps off the side (like the first one pictured in the terminology index) should work.

The tighter your chain, the harder this is going to be to snap back together.  Depends if you would prefer fiddling with a master link on a tight chain versus the re-aligning the rear hub.