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.