primary suggestion

First... this sentence is horribly written (which might be why you are having trouble parsing it).

Argue in this case is the normal sense of the verb. The "author" is making an argument.

Let's break the sentence down...

The author is arguing in support of a "primary suggestion".

The "primary suggestion" is- "fining those who do not recycle in order to pay recycling collectors more for their work."

(I am pretty sure that the use of the gerund form "fining" is grammatically incorrect.)

. . . the author is advocating for. . .

arguing in support of sth = [strongly] making the case for a certain policy to be followed.

'fining', in this situation, is not what is traditionally known as a gerund. [CGEL makes a strong case that there is no specific and separate form gerund in English]

The primary suggestion [that the author of this article argues in support of] is fining those who do not recycle in order to pay recycling collectors more for their work.

Take out the relative clause portion that I've put in brackets and we can see that 'is fining' is the verb phrase for the subject "primary suggestion".