Praise be to Allah.
It is permissible to wipe over the khuffayn (leather slippers) or socks for a reason or otherwise. The one who is not travelling may wipe over his socks for one day and one night, and the traveller may do so for three days and nights.
If a person needs to wear socks and has someone who can put them on for him, even if that is in return for payment, then he must wipe over them and it is not permissible for him to move to the option of tayammum or to regard the socks as being like a plaster cast; rather they are socks that he puts on after purifying himself completely.
It is permissible for someone else to wipe over the socks for him, as it is also permissible for someone else to help him to do wudoo’.
Ibn ‘Aabideen (may Allah have mercy on him) said in his commentary on al-Bahr ar-Raa’iq (1/182): If he tells someone to wipe over his khuffayn and he does so, that is valid as it says in al-Khulaasah.
It is not stipulated that the wiping be done directly with the hand; if he wiped with a piece of cloth or a sponge dampened with water and attached to a stick (a long-handled sponge, as you mention), that is permissible.
An-Nawawi (may Allah have mercy on him) said in al-Majmoo‘ (1/549): Our companions said: It is acceptable to wipe (over the khuffayn) with the hand, a finger, a stick, a cloth or something else.
This is also the view of the Hanbalis. See al-Insaaf, 1/160, 185.
And if he walks on a wet piece of cloth or through wet grass, that is acceptable.
It says in al-Bahr ar-Raa’iq (1/182): If the place that is to be wiped got wet by water or rain the width of three fingers, that is permissible. The same applies if he walks through grass that has been made wet by rain. End quote.
See also Asna al-Mataalib, 1/97
As you can see, the matter is quite simple, praise be to Allah.
If the disabled person does not have anyone who can wipe his socks for him or he is not able to wipe them himself with a cloth and the like, he should wash the parts of his body that are to be washed in wudoo’, and do tayammum for whatever he is unable to wash. See the answer to question no. 71202
If it is too difficult to take the socks off and put them back on again, then the ruling on a plaster cast applies and there is no time limit. So he may use them so long as they are on him, and wipe over the entire thing, top and bottom (of the feet)and to the ankles. It is not stipulated that he put them on when in a state of purity. He may wipe them himself, using his hand or something else, or someone else can wipe them for him. If he cannot find someone to wipe them for him, even in return for payment, then he may wash the parts of the body that are to be washed in wudoo’ and do tayammum for his feet, as stated above.
We ask Allah to grant well-being to all the sick Muslims.
And Allah knows best.