Compatibility issues with upgrading a 1991 1400a Miyata Road Bike
The Miyata 1400a is made of aluminum. A  7 speed Shimano 105 system uses a rear spacing of 126 mm, which is smaller than the 130 mm needed for a modern 8-10 speed system. For a steel frame, its no problem to stretching the frame and get the wheel in. On an aluminum frame, you really shouldn't. 

That being said, people have done this at their own risk, by either fidgeting with lock nuts or spacers or whatever. I wouldn't recommend it though. 

The reasonable thing to do is keep the bike as is, maybe with a different cassette if you need different gearing. You can use the 9 speed chainrings in a 7 speed drivetrain. You can also put 7 speed brifters if you want to feel more modern (you'll end up shifting more often).