Yes, it can!
Unseen Servant is a pretty smart spell!
This spell creates an invisible force that accept simple commands, and tries to perform then to the best of it's ability.
While it is, indeed, mindless, it doesn't mean it is dumb. Computers are mindless machines, and they can do a lot of stuff, including facial recognition. Mindless doesn't mean that it is "unnintellignt" - it may not be sentient, but it indeed have some sort of programming that let it do some cool stuff. You need a pretty complex cognitive processor to be able to do something like cleaning and serving food. Let's take "Cleaning" as an example.
The wizard commands the Unseen Servant to "Clean the house". That's a simple task, that a human servant definetly can do.
If someone asked here on RPG SE if I could use "Unseen Servant" to "Clean this room", I'm pretty sure the response would be yes. However, I assure you, that would be really more complex than peeking around. Let's see what would happen if we ordered it to "Clean this room".
"Clean this room" is a pretty complex task, if you stop to see what it will need to do:
First, he will check out the surroundings for something to clean. It will check if there is stuff out of place, dust on the furniture, and dirt in the floor. This means that it is intelligent enough to check if something is "dirty" or "not dirty". Alas, it is perceptive.
Then, the unseen servant will start by putting stuff that is out of place in the correct place. That may include folding clothes and puting it back on the wardrobe. Folding arbitrary clothes is a deviously complex task, that modern robots are learning to do today. Check this.
Once the stuff is back on place, the Unseen Servant will start removing dust from stuff. How would he do that? I suppose he do the same as a human servant would do: pick up a duster, and start removing dust. That means it can identify tools and USE then, which is also something that most modern robots are starting to pick up. Check this. Also, the servant must understand what "cleaning" means, which it must be smart enough to understand the difference between a clean state and a dirty state.
When the dust is removed, it will need to clean the floor. What would it do? I suppose it would grab a broom and start removing the dust from the floor. Again, perception and tool-using together to finish an objective. Also, the Servant is intelligent enough to understand when something is "done", because it knows how to stop doing something and starting to wait the commands from the caster.
So, if the Unseen Servant can do something complex as cleaning, why it wouldn't be able to peek around the corner?
Keep in mind that Unseen Servant have, in 5e, a new clause that makes it way more powerfull that it was back on 3.5: now, it can do stuff that a human servant would do. The spell descrition gives us some examples, but the limitating factor is something that a human servant would do. It can clean, cook, rub your back, mend, open door, serve food, which is another deviously complex task... heck, it could even clean your butt for you, if you are that lazy.
Of course, it telling back what it "saw" is the complicated part. You should create the order in a way that the result of the action could be easily done by the servant. I would pick up a bread, and tell my Unseen Servant to check how many people are beyond the corner and slice the bread on that many slices. If it can count how many people are around a table and serve food for then, it can definetly do that.
Alas, this kind of thing is something that can really enrich your stories. If a player is using its brains to solve problems, let he/she be creative! I really doubt that peeking around a corner could be that much game breaking.
Just remember that a Unseen Servant can't communicate back. If it was my game, I may let it be able to write, if my intent is to have a more creative game (which normally it is).
I use the following rule of thumb: If a mute, relativelly poor-skilled first level commoner could do it and it is not attack of any sort, an Unseen Servant can do it too!
Also, remember that "mindless" in this case is regarding sentience, not ability. You can have a really complex automata, complete with a giant programming that enables it to be a perfect buttler (like the A.I. Assistant from Iron Man), and it still would be a "mindless, souless machine".
Anyway, you are the DM, and the rules on 5e are more of guidelines than rules per se. You can always change stuff (and is incentivated to) if that would bring more fun for your table.