Is it obligatory for a mother to pay Zakat al-Fitr on behalf of her children who do not fast, given that their ages range between 13 and 19 years?

Children should be encouraged to fast and made to do so. As for those who have reached puberty, they must make up for the Ramadans they missed, along with repentance. As for Zakat al-Fitr, it is obligatory from the children's wealth if they have any; otherwise, it is the responsibility of whoever provides for them. It must be paid on their behalf even if they did not fast, because it is an independent obligation enjoined by the Prophet, peace and blessings be upon him, upon every Muslim.