Praise be to Allah.
Washing the feet is one of the obligatory parts of wudoo’, without which it is not valid, according to the consensus of the companions of the Prophet (peace and blessings of Allaah be upon him). This has been discussed in the answer to question no. 69761.
You can ward off hardship from yourself by doing one of two things:
1 – Pour water over your feet with a cup or your hand, etc, without lifting them up. If the water reaches the entire foot then you have washed them as required, and your purification is valid. You do not have to rub them with your hand.
Al-Nawawi (may Allaah have mercy on him) said: Our view is that rubbing the parts of the body when doing ghusl and wudoo’ is sunnah and is not obligatory, If he pours water over it and does not wipe it with his hand, or he immerses it in a large amount of water, that is sufficient and his wudoo’ and ghusl are valid. This is the view of all the scholars except Maalik and al-Muzani, who stipulated that this is a condition of ghusl and wudoo’ being valid. End quote from al-Majmoo’ (2/214)
2 – You can do wudoo’ at home and wash your feet, then put on socks. Then if you want to do wudoo’ after that, it is sufficient to wipe over the socks, for 24 hours, so long as you stay in the same city, and for 72 hours if you are travelling.
And Allaah knows best.