Can I chant without bathing, or while at work?

yes for God’s names and mental japa — they have no rules of time or place; formal Vedic or dīkṣā japa follows its own rules.

The bhakti saints insisted that the name of God may be remembered anywhere: walking, cooking, driving, lying ill in bed. In his Śikṣāṣṭaka, Śrī Caitanya says that the Lord has placed all his power in his names and set no fixed times for remembering them. Kabīr, Tukārām and Mīrā sang the same truth.

So silent or soft nāma-japa while working, commuting or doing chores is not only allowed but encouraged; it is how many people keep God in mind through a busy day.

Formal sitting japa of a Vedic mantra or a mantra received in dīkṣā is different: it is usually done after bathing or at least washing, in a clean place, seated, with the mālā. Follow the instructions of your tradition for that practice.

Source: Śikṣāṣṭaka 2 (Śrī Caitanya) · Nāma-japa: no rules of time or place

Dharma Q&A — worship at home · Ask a Jyotishi your own question