Except for the beard, which is haram to shave (obligatory precaution),[1] not only is it permissible to remove the hairs of the body such as the armpits, but it is mustahabb[2] in Islam, and has been prescribed in many hadiths[3], but isn't wajib either.
[1] Tawdihul-Masa’el (annotated by Imam Khomeini), vol. 2, pg. 835, issue 40: “Shaving the beard with a blade or machine (voluntarily) is haram according to obligatory precaution.
[2] Al-Hoseini Al-Jorjani, Al-Seyyid Al-Amir Abul-Fat’h, Tafsir Shahi, vol. 1, pg. 93, quoted by the Jame’e Fiqhe Ahle Bayt software. “It is an Islamic tradition to apply depilatory every fifteen days or forty days….”
[3] Al-Kuleini, Al-Kafi, vol. 6, pg. 505; Tahrirul-Ahkam Al-Shar’iyyah ala Madhabil-Imamiyyah, vol. 1, pg. 8 and on.