Praise be to Allah.
Firstly:
It is permissible to do tayammum when no water is available or it is not possible to use it due to sickness.
An-Nawawi (may Allah have mercy on him) said:
If water is not available, after a reasonable search for it, it is permissible for him to do tayammum, because of the verse and saheeh hadiths (to that effect), as well as scholarly consensus.
Tayammum is permissible regardless whether he is certain that he will be able to find water towards the end of the time for the prayer, or not.
This is our view and the view of all the scholars.
End quote from al-Majmoo‘ (2/301).
For more information, please see the answer to question no. 40204
Secondly:
But if it is prescribed for a person to do tayammum, he does not have to take off his shoes and it is not mustahabb for him to do that either, because tayammum has nothing to do with the feet at all.
Moreover, if he prays wearing his shoes, this is something good and is prescribed, as is confirmed by the Sunnah. The Muslim should do that sometimes, so as to establish the Sunnah and follow what is narrated therein, so long as his shoes are clean and pure, especially if that is outside the mosque or in a place where it does not matter if he wears shoes.
But in that case he should take care not to let that cause any problem with the one who is unaware of the fact that this is acceptable, or try to amaze them with that, or enter a carpeted mosque wearing his shoes were the furnishings may be made dirty by that, or the people may not understand why he is doing that, so that would cause some disturbance and confusion to them.
For more information, please see the answers to question no. 20258 and 69793
And Allah knows best.
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