You are right in determining that "the way" is the subject here
Psychologists now contend that the way adults think and feel are determined as much by their peers in early childhood than by their parents.
Anything beyond contend is modifier - "that the way adults think and feel are determined as much by their peers in early childhood than by their parents."
Looking for the subject of the modifier , you can determine that "way" is the subject

as much...as is the correct use of the idiom here and need to establish the parallel markers here which are "as much" and "as"

so , I would go with C

because
as much "by their early childhood peers" is parallel to the " by their parents"

Other examples where "and" changes the subject to plural
Humpty and Dumpty are sitting on the wall
Humpty and Dumpty had a great fall

The way" is singular. "adults think and feel" is just used to modify/quality "the way" The verb form should be singular.
Eliminate A,B, and E.

or just to determined what subject is ask question what determined think and feel - you get the answer the way, which is singular ..

What the psychologists contend should use general state form of verb.

Eliminate D.

Hope it helps

+1 for C.
The way is the subject here