Did Swami Vivekananda regularly chant any Vedic or Tantric mantras during his lifetime?
Yes, he did. what they were specifically, is not said in the books. Vivekananda become proficient in Sanskrit grammar at a young age and spent time. He was able to speak and do Sanskrit argumentation with Sanskrit scholars. He believed and said that all Indians should learn Sanskrit.
It is said that he would sometimes spend many hours in Japa and meditation. In the Complete Works (V7, p 62) Vivekananda says:
Repeating the Om continually is the only true worship. It is not a word, it is God Himself.
Based on this one teaching, you can be sure he repeated AUM regularly.
Yes, Sri Ramakrishna surrendered Swamiji to Maa Kali and asked him to meditate on the form of Maa Kali. So He had given Kali Mantra to Swamiji in all probabilities.Swamiji soon started getting visions of Kali, different other deities and mantras, as He told confidentially to Sri M in 1887(as recorded in Sri Ramakrishna Kathamrita.
Sri Ramakrishna also gave Him Rama Mantra at Kashipore in 1886 (as mentioned in Sri Ramakrishner Antyaleela by Swami Prabhananda).He was taught Swamiji Advaitavaad as mentioned by Swamiji in His Complete Works.
It seemed Swamiji, besides chanting OM , worshiped Lord Shiva and Maa Kali frequently.He told Nivedita that he regularly worshiped Kali and mentioned it as his personal affair which he was not ready to preach.
Normally incarnations like him can not do japa for long.They soon get immersed in meditation, bhava and samadhi, as happened with Sri Ramakrishna.
Chanting of the Raama Naama was also very favorite of Swamiji.