Is it Islamically correct for a mother to perform 'Umrah despite her severe illness and inability to walk, insisting on it even though her children advised her to donate money instead?

Umrah is obligatory upon the mukallaf (accountable individual) who is able, either by performing it themselves or by appointing a proxy. If the mother is financially capable but physically unable, she must appoint someone who has already performed Umrah for themselves. Alternatively, she can go herself and perform tawaf and sa'i while riding in a wheelchair or similar conveyance. Prioritizing Umrah over charity is obligatory.