If so, it is used to describe someone who is easy going/ easily led- a tappy larry would drag along at the back of a group, happy to do what everyone else was doing.
A Geordie folk-club performer years ago used to tell a shaggy-dog story about an ancient historical character - I can't remember who now, and it's not really important - which involved this character "and all his cuddies (donkeys or ponies), and all his cuddy-chariots, and all his cuddy-chariot drivers, gannin' tappy-lappy doon the moontainside", so I guess it means "as fast as they could manage".
I can't remember that either - after all, it was in the 60s! But the phrase just stuck in my head. I believe said performer is still on the scene, so I could probably get the story with a bit of effort..........:-)
When I've heard 'tappy lappy' used it has always been in the context of doing something quickly, or at speed. 'Y'best get ower there tappy lappy,' You can also imagine a criminal 'gannin tappy lappy t'get away from the poliss.' So I would think 'tappy lappy' is a term meaning 'faster than normal.'