More than 11,000 ballots are thought to have been taken in the state of Tabasco on Monday.
A truck blocked a highway and halted the journey of staff from the National Electoral Institute.
A group of armed civilians are said to have ransacked the vehicle and stolen packets of voting papers.
More than 8,000 ballots were also taken and burned in the state of Oaxaca on Tuesday, according to local media.
Electoral officials said ballot papers were being delivered when the vehicle was intercepted by unknown assailants.
All voting officials were unharmed, but in total 8,204 voting papers were destroyed.
The Oaxaca Electoral Institute confirmed it planned to reprint the destroyed voting papers.
In a statement they said they would carry out “the steps and actions necessary so that the burned ballots are reprinted” in time for the weekend’s vote.
There is a chance around 100 polling stations will be closed for the July 1 vote.
Electoral authorities are having problems installing the voting stations in some areas of the country.
Marco Antonio Baños, an electoral councillor said: “The number of voting booths with problems related to their installation has increased but it’s not something that places any of the elections at risk.
“We’re talking about around 100 booths out of 157,000” election stations that will be open.
Violence has overshadowed the election so far.
More than 100 politicians have been killed since September, and more than 13,000 people have been murdered in Mexico since the start of this year.
More than 18,000 positions are up for grabs across the country in Sunday’s vote.
Corruption, security and relations with Trump’s America have all been key issues in the run up to this election.
At the presidential level, former Mexico City Mayor López Obrador has come out as the frontrunner, with the latest polls showing he has the support of 51 to 54 percent of the electorate, a lead of more than 20 percent, according to Reuters.