All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
It is permissible for the Muslim to recite the Quran while standing or walking, for the Prophet, sallallaahu ‘alayhi wa sallam, used to remember Allaah The Almighty at all times, as previously clarified in Fatwa 86104.
When Mu‘aath asked Abu Moosa Al-Ash‘ari how he recited the Quran, he replied: "I recite it while I am standing, sitting or riding my riding animals, at intervals and piecemeal." [Al-Bukhari]
However, some Muslim jurists held the opinion that it is disliked for the Muslim to recite the Quran while walking if his heart would not reflect on its meanings, and also while walking in the market, but there is no evidence that this is dislikeable, since reciting the Quran is a form of Thikr, and Thikr is permissible in such places.
Allaah Knows best.