I hadn't driven my Datsun in a while and I need to move it. When I went to start it, at cranking, there was a very bad vibration, like it seemed like the whole engine was shaking.

Didn't really want to start it with it behaving like that but I really needed to move the car, so I started it anyway. Once it was going with the clutch engaged it seemed better.

The car was fine when I last drove it. I am thinking the flywheel might have started to come loose on that last drive.

Should I be able to confirm this without having the transmission out ? Could I say just pull the starter and have someone press the clutch while I try and wobble the flywheel ?

Any other ideas ?

I have the starter out now and I can rotate the flywheel back and forth

Do you mean that the flywheel rotates a bit, but the engine font pulley doesn't move with it? Loose flywheel bolts are not that common unless the engine is revved over 7,000 rpm or so. When you fix it, just make sure the bolts are torqued to specification