He cannot stand to do ghusl because he is sick; can he do tayammum?

Firstly:

If someone has a problem or becomes sick, then he must do the obligatory duties to the best of his ability, and should not burden himself with more than he can bear.

Allah, may He be exalted, says (interpretation of the meaning):

{Allah does not burden any soul with more than it can bear} [al-Baqarah 2:286]

{So fear Allah as much as you are able} [al-Taghabun 64:16].

It was narrated from Abu Hurayrah that the Prophet (blessings and peace of Allah be upon him) said: “If I forbid you to do a thing, then avoid it, and if I instruct you to do a thing, then do of it as much as you are able to.” Narrated by al-Bukhari (7288) and Muslim (1337).

So the one who is sick should do whatever he can of the obligatory parts of purifying himself and praying, and whatever he is unable to do is waived in his case.

Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:

Whoever studies what is mentioned in the Qur’an and Sunnah will realise that accountability depends on the ability to learn and act. If someone is unable to do either of the two, what he is unable to do is waived in his case, for Allah does not burden any soul with more than it can bear. … Hence the Prophet (blessings and peace of Allah be upon him) said to ‘Imran ibn Husayn: “Pray standing; and if you cannot, then (pray) sitting; and if you cannot, then (pray) lying on your side.” This is an important guideline."(Majmu‘ al-Fatawa  21/634).             

What we understand from your question is that you are only unable to stand. Therefore you must try to do ghusl in the manner in which you are able to do it, by doing ghusl whilst sitting on a chair, or in a bathtub, because it is not one of the conditions of ghusl that you must be standing.

It is not valid to content yourself with tayammum unless you are unable to do ghusl at all. In that case, it is valid for you to do tayammum.

Ibn al-Qattan (may Allah have mercy on him) said:

The scholars are agreed that if a sick person will be harmed by using water, even though he can find it, then he may do tayammum."(Al-Iqna‘ 1/92).

For more information, please see the answer to question no. 105356 .

Secondly:

Tayammum in the case of janabah is the same as tayammum instead of wudu’.

Allah, may He be exalted, says (interpretation of the meaning):

{O you who have believed, when you rise to [perform] prayer, wash your faces and your forearms to the elbows and wipe over your heads and wash your feet to the ankles. And if you are in a state of janabah, then purify yourselves. But if you are ill or on a journey or one of you comes from the place of relieving himself or you have contacted women and do not find water, then seek clean earth and wipe over your faces and hands with it. Allah does not intend to make difficulty for you, but He intends to purify you and complete His favor upon you that you may be grateful} [al-Ma’idah 5:6].

It was narrated that ‘Amir ibn Yasir (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) sent me on an errand, and I became junub, so I rolled in the dust like an animal. I mentioned that to the Prophet (blessings and peace of Allah be upon him) and he said: “Rather it would have sufficed you to do like this,” and he struck his palms once on the ground, then he shook off the dust, then wiped the back of his [right] hand with his left hand and the back of his left hand with his [right] hand, then he wiped his face with them [his hands]. Narrated by al-Bukhari (347) and Muslim (368).

For more information on how tayammum is to be done, please see the answer to question no. 21074 .

Thirdly:

If one who is sick prays after doing tayammum because he is not able to use water, then his prayer is valid, because the one who does an act of worship in the correct manner should not be told to repeat it.

Rather when he recovers and becomes able to do it, he should do ghusl for the upcoming prayers.

Ibn al-Mundhir (may Allah have mercy on him) said:

The scholars are unanimously agreed that if someone does tayammum with clean earth, as Allah has enjoined, and prays, then he finds water after the time for that prayer has ended, he does not have to repeat the prayer."(Al-Awsat  2/63).

And Allah knows best.