It would be possible if there was a factory turbo version of your engine ever offered but there was not.
Say you wanted to use a turbo set up from a wrecked subaru. You would need the Turbo manifold (exhaust mainfold with a flange that bolts to a turbo), but the manifold would not bolt up to your engine because they are different. Also the piping to the intercooler would not fit around your engine. Also your ECU/air flow sensors/fuel system/enginge internals are not designed to support boost. You get the idea.
Say you wanted to pull all the supercharger parts off of a Gen1 MR2 with the 4a-gze. The manifold and piping would bolt up, but your 4a-ge doesn't have semi-forged pistons and lowerd compression like the gze, and your ecu and fuel cannot spport boost. If you wanted to do an engine swap you would have to contend with the conversion from MR to FF, etc.
Custom turbos and engine swaps can be done, but to start you will definitley need to do a lot of research on how your car works and what options are out there. Try reading some of the stuff here in the master corolla sticky and check out the tech section over at www.club4ag.com.
Also check out some of the set-ups at www.turbo-corolla.com