Tiruvashtakshara japam is done during sandhyavandanam. It can also be done during various other special occasions as well. Women do Tiruvashtakshara japan instead of sandhyavandanam. The details can be found in the previous issues of Sudarsanam.
Saints have several other mantras, including many Veda mantras, other than Tiruvashtakshara mantram for the penance. Veda parayanam itself is knows as a japam as per a Vedic statement. The Rishis through their constant recitation of Veda mantras have safeguarded the Vedas for us. There is no separate book for the Vedas. Only through the japam of the Rishis, we have obtained Rig, Yajur, and Sama Vedas; they used to wander around forests continuously chanting the Vedic mantras; such kinds of practices have ensured we have access to the Vedas.