Police are investigating the attack but have so far found no suspects.
Jha was elected as Nepal's first vice-president in July after the Himalayan nation abolished its 239-year-old monarchy.
He sparked controversy by taking his oath of office in the Hindi language, identified with Nepal's Indian heritage, rather than in Nepali.
His critics demanded his resignation, but protests died down after he apologised.
Jha, an ethnic Madhesi, represents the Madhesi People's Rights Forum, which has called for greater rights for the Madhesi people of southern Nepal and was behind protests at the beinning of the year.