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She is deeply ashamed that her son committed theft.
Literal
She [topic-は] son [subject-が] theft [object-を] worked [genitive-の][because-of-こと-で] terribly shameful experience [object-を] is-having.
盗みを働く ('commit theft, do stealing') is a fixed verb-noun collocation: 働く ('work, function') here in its 'commit / engage in (an act of)' sense — paired with criminal nouns like 盗み ('theft'), 詐欺 ('fraud'), 悪事 ('wrongdoing'). ~ことで marks the cause/grounds ('because of the fact that X'). 恥ずかしい思いをする ('have an embarrassing/shameful experience') uses the productive [adj]+思いをする 'have an X experience' frame — same pattern as 嫌な思いをする, 悲しい思いをする.