Virat Kohli-led Royal Challengers Bangalore had a below-par IPL 2019 and failed to make it to the playoffs.
Lehmann had earlier coached Brisbane Heat in 2012-13 season.
Vettori has been coach since 2015 and will end his tenure when the season finishes.
The decision led to chaos on the field with Heat coach Daniel Vettori and skipper Chris Lynn engaged in animated discussions with the officials near the boundary rope.
Mujeeb, 17, has already played 23 ODIs, seven T20Is and Afghanistan's inaugural Test match.
Left-arm paceman Neil Wagner became the eighth New Zealand bowler to complete 150 Test wickets
The 17-year-old has joined the franchise as its second overseas signing after Brendon McCullum.
Former South Africa allrounder Lance Klusener has been named as the head coach of the Bangladesh Premier League franchise Rajshahi Kings.
Head coach Vettori, bowling coach McDonald, and batting and fielding coach Woodhill are all on the verge of being sacked.
After the World Cup in England, New Zealand will play a total of 28 international matches leading up to the conclusion of the 2020 World T20 in Australia.