Can children apply to the Pension Fund of Ukraine for a pension on behalf of a pensioner?

Yes, they can. 

An application for a pension may be submitted by a representative, including a relative of the pensioner, who is acting based on a notarized power of attorney. 

In addition, in the interests of a pensioner, based on a notarized power of attorney, it is possible to apply with other related applications, for example, with an application for pension recalculation, termination of charge to a bank account, transfer from one type of pension to another, etc.