Is it permissible for the son's wife to care for her elderly father-in-law when his 'awrah is occasionally exposed, in the absence of her husband?

If your father is in need of marriage and you are able to help him marry, then that is your duty. If your father gets married, it is not permissible for your wife or other of his mahārim (unmarriageable kin) to look at his nakedness (awrah). If it is difficult for him to marry, or if his wife does not take care of him, then men are more deserving of taking charge of him. If that is not possible, then there is no objection to some of his mahārim taking care of him, and your wife is among them.

Your wife must restrain herself from looking at his nakedness when it is exposed. If she needs to clean him and remove impurities from him, she should lower her gaze as much as possible and not touch his private parts directly, but rather use a cloth or something similar.