The correct option is C Might
Modal verbs help determine the mood of a sentence, to indicate obligation, permission, possibility and so on. The underlined modal verb "may" indicates a possibility of the person "she" joining "us" for dinner, which is true. The sentence does indicate a possibility, but, 'may' isn't the correct modal, because it's used in direct speech. 'She said, "I may join you all for dinner"' is in the direct speech, whereas the given sentence is in indirect speech. 'Might' is used as past tense of 'may' in indirect speech (while changing the speech from direct to indirect, the tense is changed to past tense), hence option A is incorrect and C is correct. 'Can' indicates ability, like in 'She can speak English very well', here it's not very suitable because the sentence requires a modal that explains that the person "she" may or may not join in; 'could' is used as past tense of 'can' in the indirect speech. Since the sentence does not focus on ability, but possibility, options B and D are incorrect.