Shimano STI brifters are not servicable. Shimano does not sell repair parts or publish a repair manual. There are a few guys who claim they have repaired various versions of STI's but there is no commercial service for this I know about. If you decide to go inside one, you are pretty much on your own.
The usual "fix" is to hose out the mechanism with copious quantities of WD-40 to wash out the accumulated crud and old lube and then lube them with light oil like TriFlow. This will often get them working again. Otherwise they have to be replaced.
Campy's Ergo brifters are servicable and both parts, instructions and commercial rebuilding services are available. Shimano has none of the above.