The Italian has called for "sanctions" against the two teams after the Dutch squad withdrew from the race following Steven Kruijswijk's positive COVID-19 test on the first rest day, while EF wrote a letter calling on organisers RCS Sport and the UCI to stop the race at the second rest day.
During the race, Vegni defended the race's anti-COVID-19 bubble after EF had called it "clearly compromised" following 11 positive tests across four teams in the race, including those of Kruijwsijk, Simon Yates, and Michael Matthews.
EF called for additional COVID-19 testing as well as additional measures to reinforce the bubble, both of which were put in place by the UCI and RCS Sport hours later. Jumbo-Visma, meanwhile, pulled out of the race on the morning of stage 10, citing a desire to protect their riders and staff.
Vegni had approved of Mitchelton-Scott's decision to withdraw from the race following five positive tests, including Yates', but followed EF's letter by hinting at "incorrect behaviours" by teams within the bubble.
In an interview with La Gazzetta dello Sport published on Tuesday, after the Giro had finished in MIlan, Vegni stated that he wants "sanctions" for both Jumbo-Visma and EF.
EF Pro Cycling call for Giro d'Italia to be stopped
Vegni defends Giro d'Italia COVID-19 bubble after EF Pro Cycling call for race to stop early
Jumbo-Visma pull out of Giro d'Italia to protect riders and staff after COVID-19 outbreak
Vegni: 'Somebody will pay' for protest that shortened stage 19 of Giro d'Italia
"I said that there'd be a second part of the Giro," he said. "And it will be discussed where it should be, to make the right decisions; I'm thinking in particular of the UCI. We will ask those in charge what they intend to do. It doesn't end here.
"I want sanctions for Jumbo-Visma and EF. If anyone thinks they can infringe the rights or duties of WorldTour teams, then I will also consider myself 'free' from the obligation to invite them. What happened is also a matter for the UCI License Commission."
EF Pro Cycling and Jumbo-Visma both hold WorldTour licenses valid for the period 2020-2022, so it is unclear which of the four licensing criteria – ethical, financial, administrative and organisational – though per article 2.1.5.032 of UCI regulations, the License Commission may reduce the duration of validity of a license should a licensee break with the four criteria.
The UCI's Professional Cycling Council may also suspend a team's license "if this is considered necessary to protect the image of the WorldTour." To date, neither step has been taken against any WorldTour team.
Vegni also called for a new rider's union following the events of stage 19 in Morbegno, which saw CPA representatives vote for a stage reduction, followed by riders successfully agitating for the stage to be reduced at the start.
"We need a different mentality and a different union," Vegni suggested.
"The riders must have a union that helps them and that doesn't just fight to the bitter end. The riders are intelligent people and there's a need to change the idea of the union, for it to modernise, so that looks at the real interests of the riders. Fights should be fought in a meeting room, not in the streets.
"The fact that they sent a rider representative during his last race [Adam Hansen of Lotto Soudal] says it all. The maglia rosa Kelderman, who supported the protest on TV, later apologised to me in Sestriere."
On the day of the stage, Vegni had threatened the union and its representatives if there was no proof of a vote to shorten the stage, saying "someone will pay." He did not mention the proposed legal action during the Gazzetta dello Sport interview.
For their part, the CPA released an open letter on Monday, detailing the reasons for shortening stage 19 while also asking for growth in terms of rider safety.
"All of [the races] are precious for us and for the whole movement, of which we are the most exposed actors, for better or for worse. We deserve to be listened to, even when we say something you don't like," the letter concluded.
Vegni said he was proud that the Giro d'Italia made it to the finish in Milan despite the COVID-19 pandemic and the tensions during the race.
"Someone perhaps thought we wouldn't have made it but we did and with a time trial that decided the race," he said.
"The Giro sent out a message of pride for all of Italy. It's a success for the country and a sign that it is possible to live with the virus.
"I've got to thank all my staff, who worked so hard. This year feels like ten and it's definitely the most difficult race I've ever organised. There were lots of worries and lots of things to resolve but we always did it with the desire to continue and to reach Milan."