From the Oxford Dictionary of English Surnames:

The name Walker is actually derived from Fuller--Old English: fullere, 'a fuller of cloth'. The raw cloth has to be fulled, i.e. scoured and thickened by beating it in water, a process known as walking because originally done by men trampling upon it in a trough. Hence Walker, by the side of Fuller and Tucker from OE tuchan, originally to torment, later to tuck, to full. These three surnames seem to be characteristic of different part of England. In general, in Middle English, Fuller is southern and eastern; Walker belongs to the west and north, Tucker is south-western.