。
She has an automatic washing machine.
Literal
She [topic-は] automatic [possessive-の] washing-machine [object-を] has.
自動式 ('automatic-style, automatic-type') uses the productive suffix ~式 ('style, type, system'), which appears across countless compounds: 旧式 ('old-style'), 西洋式 ('Western-style'), 接触式 ('contact-type'). The の linking 自動式 to 洗濯機 is the standard noun-modifying の. The wording of this sentence has a slightly dated feel — 'automatic washing machine' was a meaningful distinction in postwar Japan when twin-tub manual machines were still common, but the 自動式 qualifier is now redundant since virtually all washers are automatic.