Is it possible to go war with somebody with a fairly limited army (plenty of turrets, some tanks and troops, no jets), so all attacks would be done with spies? Essentially, the traditional army would be for defense only and all acts of aggression would be done through covert operations.
How well would this work, and what would be a few good strategies to accomplish a win?

it'd fail, limited spyops per day and target would recover quicklish and gdi could prevent spyattacks.

then what would be the most effective way of using my spies in combat scenarios?

bio, stir and cd.

FAIL

You can annoy them with spy ops, but you won't kill them.

If you are killing then Stir Rebs helps greatly, particularly on express as turns regaining readiness are still reducing population.

If you want to create a parking lot Bomb Buildings are useful as well as turns regaining readiness are still destroying buildings.