What is the word for act of monotonously demanding for something (many be a toy) repeatedly?

Both nagging and pestering would fit.

nag: to annoy by persistent faultfinding, complaints, or demands.

pester: trouble or annoy (someone) with frequent or persistent requests or interruptions

And there is a phenomenon called "pester power":

the ability possessed by a child to nag a parent relentlessly until the parent succumbs and agrees to the child's request

The verb is to pester:

pester : (transitive) To bother, harass or annoy persistently.

A mother might say to her child, "If you stop pestering me about it, I might consider it."

A more protracted and calculated persistent request for something might be called a campaign

campaign : a connected series of operations designed to bring about a particular result

Suzy has been campaigning to get that doll for two months straight.