Putting Shimano SORA STI Shifters on Touring Bike with Deore Derailleur
For the rear, a 9 speed Sora brifter will be fine -- 9 speed cable pull is the same between Shimano road and mountain.

For the front, the bike has a Shimano Altus front derailleur. You'll need to swap a road triple front derailleur for that (make sure to check the spec sheet to ensure your crankset's chainring sizes are within spec (not a problem) and the difference between the largest and smallest chainring are within spec (can be a problem) ) -- the bike you have has a tooth difference of 22t for the crankset, which is slightly higher than the 20t that the Sora FD-3403 or FD-3503 is rated for. 

There are triple front derailleurs (e.g. Microshift R539) which will have the right front tooth difference though (comes in braze-on and band mounting, depending on the type of mount on your frame).