Sora STI Shifters with Deore
There are many answers here that cover variations of this question in more depth, but the answer is not really, because Shimano front derailer cable pull is different between mountain and road. What tends to happen as a result of the combination you describe is it does shift, but performance is lacking and there will invariably be FD cage rub in some gear combinations  that you'll never be able to adjust out. You read conflicting things about this because it does in fact shift and kinda sorta work from a certain perspective and that's good enough for some people. The rear is matched perfectly because all 9 speed Shimano shifters are the same cable pull, mountain or road.

Solutions include bar-ends with their friction front shifting (which is good for other reasons on touring bikes because it's less finicky and tends to work better in situations where you've chosen whatever chainring sizes you want as opposed to what's prescribed), a shift adapter such as Jtek, or pushing the limits of what road front triples can do (this also compromises performance). Another very good approach is using 10-speed pre-Escape Ergo levers with the so-called Hubbub mod aka Shimergo, which is a hack that makes them work with Shimano 9-speed rear derailers and cassettes while the front has lots of small clicks and can basically shift anything. In practical terms most people just use bar-ends here and leave it at that though.