Lesufi, who is also MEC for basic education in Gauteng, said activists, including himself, were willing to lobby the government and government institutions to cancel contracts associated with Vodacom.
The planned rolling mass action is also aimed at isolating Vodacom internationally.
"We are starting discussions with international organisations, that we know are pro-customers, to assist us to isolate Vodacom and its brand internationally.
"We are going to formally register our displeasure and dissatisfaction on how Vodacom is handling this matter. We believe this matter deserves their attention and we ask for their support," he said.
Lefusi also called on all Vodacom customers "to request an airtime and data advance and refuse to pay so that Vodacom can taste its own medicine".
He said Vodacom subjected Makate to "a serious commercial injustice" and called on all South Africans, across political lines, social class, labour organisations and youth movements to join in the boycott.