Thanks @InThrustWeTrust for tagging me on this one. I always try to help anyone who is looking to do custom work and modifications. That someone with the knowledge you have would tag me is really cool, so thanks again.

To the question at hand: the short answer is yes, it can be done but you are honestly going to spend more money doing it than you should. The Yaris was not designed with forced induction in mind, and if you want to add 25% more power you're going to be pushing her hard. Most guys engine swap the Yaris because it is actually cheaper than going FI. If you are determined to do it, there are only two real options out there: Zage makes a turbo kit for it, and Blitz Racing does a supercharger. 

Before we go any further, it is worth asking how much you are willing to spend on this. These kits are low production and usually have to come from Japan. You are looking at $6000 just in parts for either option, PARTS ONLY. And that does NOT include very important, mandatory items such as tuning, a new ECU, wiring harnesses, fuel system improvements, better engine internals, a transmission that could handle the power, and more. In all honesty, you would be in the $12k to $15k ballpark at a minimum. And even then, there are constant issues with tuning and making the car run right. It would not be suitable for daily driving and reliability would be extremely iffy. And that's not because a shop can't build it right, we can. But you are talking about upping compression significantly on a very small motor. 

Toyota themselves released a supercharged Yaris variant in Europe called the Yaris GRMN. And even their gurus, who would have access to data nobody else would, could only get 205 horsepower out of it. It also ruined the ride as the car was mostly built for rally racing and reliability was pretty much set aside.

All told, I wouldn't recommend it myself and I wouldn't do the job at my shop. Just too much risk of popping the motor trying to squeeze out another 25%. I can direct you to a thread about this topic on a Yaris forum though:

https://yarisworld.com/forums/showthread.php?t=58452

As you can see, it has been much discussed but almost everyone goes engine swap which is a very expensive route as well, assuming you can even source a good candidate engine. That site is for Yaris enthusiasts and even they have mostly passed on the idea. And of those that did try, none have dependable cars anymore.

Sorry if too long of a reply, but you are talking a major project. I'm not being at all antagonistic when I say you could keep your Yaris and buy a faster car to go with it for less than it would cost you to do forced induction on the Yaris. It just isn't a good plan, and I'm sorry to be the bearer of bad news. 

Almost forgot: yes, to get what you want, you would definitely be running premium gas. And if you are AT ALL concerned about gas prices, do not even think about attempting something like this. Just being honest and truthful, no disrespect intended in any way.

Honestly, you can do whatever you wish. Just realise that once you change the stock setup, you can kiss your *ultra reliable little car* goodbye. 

If you’re fine with that, sure, just use a good shop who knows their stuff for the job. 

@Mod_Man may be able to tell you the pros & cons of what you’re about to do, better. 

Do you really mean supercharging? Or turbocharging? Sometimes it's used interchangeably...

Boosted engines need higher octane to prevent pre-ignition, so yes you would need to use premium gas once you supercharge your engine. Please don't ask if you can use regular gas to save money...we've explained the importance of octane on multiple topics which you can look up. 

As for supercharging the engine, I guess you can if you really wanted to. In my opinion, you should save your money towards a more performance oriented car if you want more power under your feet. You're gonna put money towards a supercharger kit that will wear out your engine faster without the proper upgraded internals. You didn't state how many miles were on the vehicle as well, and that will play a role in how long it will last too with the added boost. But if you do really want to supercharge the engine, the MAX boost I would run is 6-7 psi...anything more will just ask for trouble. Also, make sure you do a compression test and leak-down test before even buying one to make sure your engine is healthy enough for the added boost and strain the supercharger will provide. After all, installing a supercharger on an already bad engine will just blow it up within a few hundred miles (and it's not fun, I've seen it happen).

To me it sounds like it's the wrong car for you.

1. Yes I did mean supercharging, because a turbo is even harder to add, and I don't want turbo lag. The existing performance of the car is quite adequate but it would be so fun to have a stealthy rocket. But the cost does seem unreasonable.