Solution:(d) The old woman requested her neighbour to buy some vegetables for her the next day. (Correct)(a) The old woman requested to her neighbour to please buy some vegetables for her tomorrow. ("Please "is superfluous and tomorrow is incorrect)
(b) The old woman told her neighbour that buy some vegetables for me tomorrow, (That and tomorrow are incorrect)
(c) The old woman ordered her neighbour to buy some vegetables for me the next day. ("Me" in the reported speech is Incorrect)
Note- The word 'Please' shows some sense of request. So, 'please' in the direct form converts to 'requested' in the indirect form. Further, 'please' and 'requested' cannot come together in a sentence in the indirect form. Going through the options, only option d satisfies this condition.