Yes, you'll be able to run 9sp Sora shifters in place of 9sp Tiagra ones.
As long as you're sticking to Shimano (keeping it simple, becomes more complex if you want to mix Shim, Campag and SRAM so we'll not go there, except that SRAM cassettes are interchangeable with Shimano ones, but the shifters and mechs aren't), then the only things that care are the shifters and the cassette. they have to be the same number of speeds.
The chain should be the same as the cassette, or 'more speeds' (i.e. narrower).
The front & rear derailleurs and the chainset don't care at all, although it's easier to set-up if all the 'same speeds'.
My TT bike for instance has 8sp Ultegra bar-end shifters, 8sp Shimano cassette, 9sp Ultegra front mech, 10sp 105 rear mech, 9sp SRAM or KMC (I forget) chain, 9sp 105 chainset
- the shifters and cassette match, the rest just works.