No, people with small eyes will not see less than people with big eyes. Although the range of the lights coming into the eye balls may be a little different for people with big eyes and small eyes, they will see almost the same to the outside world. Eyes are the important parts for our whole body. We need to protect them carefully.
It is not very possible that people with small eyes see smaller area than people with big eyes. This is because that it is the size of the pupil of eyes which matters, and that stays the same no matter the eyes are small or big. The iris and sclera will change when the eyes get bigger, but they don't really play a part in how much of the area people can see. In general, the size of eyes can affect the eye vision to a certain degree, however, it is definitely not the only component that determines the vision of people. There are in the structure of human eyes two parts are responsible for the vision of each indivisual, that is, the power of the cornea and the length of the eyeball. If the eye of an indivisual is too long and the cornea cannot focus light on the retina, it can lead to the nearsightedness. On the other hand, if the eye is short and the cornea cannot focus light on the retina which will lead to the light focusing behind the retina, then the lens will try to help by adding some focusing power, which may or may not focus the image clearly causing farsightedness. In addition, there is a kind of condition called microphthalmos which refers to those people who have got a disorder of smaller eyes with flat and not very powerful corneas, thus leading to highly nearsighted eyes.
As far as i am concerned, the area we see have nothing to do with whether our eyes are big or small. In reality, it is related to the radian of our crystalline lens essentially. Therefore, the size of our eyes can not decide the scope of the area we see. Hope it can be helped.