わりに and わりと are not always interchangeable because they are slightly different in terms of the nuance, though they both signify 逆接.
わりに emphasizes some property of the subject, using another property that usually affects the said property in an adverse manner.
Take an example:
彼は毎日ラーメンを食べるわりに健康だ。
In this sentence, the health of the subject is emphasized by presenting the fact that he eats ramen every day, which usually has a negative effect on the health.
This is sometimes incompatible with わりと, which emphasizes some property comparing the subject to the average.
Take an example:
彼は50代にしてはわりと若く見える。
In this sentence, the subject is described to look younger than an average person in his/her 50s.
One can use わりに in this case:
彼は50代のわりに若く見える。
The replacement is possible because "the group from which the average person is chosen" is explicitly present.
The average person in his/her 50s has an adverse property, namely the age, that affects the appearance in an adverse manner.
The sentence in question:
たくさん食べる人普通は、わりと体が大きいですよね。
does not state to which average the physique of the person should be compared.
In such a case the use of わりに is not possible because you do not know which adverse property to present.
As for your second question, わり in わりと should not be confused with the noun わり(proportion).
They are simply different words, although I believe they have similar origins.