Injured players Trent Boult and Corey Anderson will miss the tour, as will skipper Brendon McCullum and paceman Tim Southee, who are both being rested.
Kane Williamson will captain the side in McCullum's absence.
Sodhi, who previously represented New Zealand at test and Twenty20 level, is in line to make his ODI debut for the Black Caps.
He was one of three players to miss out when New Zealand Cricket last week named its 20-strong list of contracted players.
Batsman Colin Munro, another to miss out on a contract, has also been included in the team which will play six ODIs and three Twenty20 matches over 25 days beginning in Harare on August 2.
Sodhi will be replaced by seamer Doug Bracewell for the South African leg of the tour, which begins in Durban on August 14.
Black Caps coach and selector Mike Hesson says the choice of Sodhi for Zimbabwe and Bracewell for South Africa was very much a matter of picking horses for courses.
"We think Ish will be useful on pitches which tend to be slow and take turn, and against a Zimbabwe team stacked with right-handers," he said.
"But we're unlikely to need a third spinner in South Africa, which is why we've opted for another seamer in Doug."
Hesson said the absence of injured and rested players provided plenty of opportunity for a host of returning players to further their international credentials.
Batsmen Tom Latham and Munro are joined by pace bowler Adam Milne and allrounder James Neesham, who return to the fray following layoffs due to injury.
"There are chances there for batsmen such as Tom and Colin, for Ish and for a handful of bowlers," Hesson said.
"And, not least, it's another good opportunity for (stand-in captain) Kane Williamson to continue upskilling in terms of leadership."
New Zealand: Kane Williamson (captain), Doug Bracewell (South Africa series), Grant Elliott, Matt Henry, Martin Guptill, Tom Latham, Mitchell McClenaghan, Nathan McCullum, Adam Milne, Colin Munro, James Neesham, Luke Ronchi, Mitchell Santner, Ish Sodhi (Zimbabwe series), Ross Taylor, Ben Wheeler.
