His wife cannot do ghusl because of sickness; can he have intercourse with her and let her do tayammum?

There is nothing wrong with you being intimate with your wife, even if the matter is as you describe, and she cannot use water because she is sick. It is sufficient for her to do tayammum until Allaah heals her and she can do ghusl. 

Abu Dawood (333) narrated that Abu Dharr (may Allaah be pleased with him) said: I was staying in a place that was far from any source of water and my wife was with me. I became junub and I prayed without purifying myself (i.e., without doing tayammum or ghusl). I came to the Messenger of Allaah (peace and blessings of Allaah be upon him) and he told me to use water and do ghusl. Then he said: “Clean earth is a means of purification even you cannot find water for ten years.” Classed as saheeh by al-Albaani in Saheeh Abi Dawood. 

Some scholars regarded it as makrooh for a man to have intercourse with his wife when he does not have water with which to do ghusl. But the correct view is that it is not makrooh, because of the hadeeth of Abu Dharr quoted above. The Prophet (peace and blessings of Allaah be upon him) did not tell him not to do that, rather he told him that tayammum is sufficient until he finds water and can do ghusl. 

Ibn Qudaamah (may Allaah have mercy on him) said, after quoting the view of those who regard that as makrooh: 

It is more likely that it is permissible to have intercourse with her and it is not makrooh, because Abu Dharr said: The Prophet (peace and blessings of Allaah be upon him) said: … then he quoted the same hadeeth.  Ibn ‘Abbaas had intercourse with a slave woman of his, when he did not have any water available, and he led his companions in prayer, among whom was ‘Ammaar, and they did not criticize him. Ishaaq ibn Raahawayh said: This is a Sunnah which is narrated from the Prophet (peace and blessings of Allaah be upon him) concerning Abu Dharr, ‘Ammaar and others. End quote from al-Mughni (1/171). 

And Allaah knows best.