The Halfords store closed in August this year, with customers directed to the chain's Brixton store instead.
The Tapi chain was set up in 2015 by Martin Harris, son of Carpetright founder Lord Harris of Peckham.
The new Tapi shop will compete with Carpetright - sold by the Harris family a decade ago - which has an Old Kent Road store on the Southernwood Retail Park near Tesco.
The currently empty ex-Halfords store is part of the Cantium Retail Park which changed hands a year ago for £38 million.
The previous owner of the retail park - which also includes the B&Q and Pets at Home stores - won approval from Southwark Council in 2019 for a development of 1,100 homes in towers up to 48 storeys high.
This month a deputy mayor of London warned that the redevelopment of the Old Kent Road is "stalling" without confirmed funding for the Bakerloo line extension.
Cantium Retail Park's current owners - LondonMetric - are in the process of carrying out some small-scale works in the car park which will 'implement' the 2019 planning permission for the skyscraper development to prevent it from expiring.