'waddle' (source of 'waddling') originates from Middle English 'wadelen', a frequentative form related to 'wade' (to go in or to move), ultimately connected to Old English 'wadan' meaning 'to go, move'.

'waddling' developed from Middle English 'wadelen' (a frequentative of 'wade') and through Early Modern English became the modern verb 'waddle' with the participle 'waddling'.

Initially, the root meant 'to go or move' in a general sense; over time the frequentative form came to mean specifically 'to walk with short, rocking steps', which is the modern sense of 'waddle' and 'waddling'.