In the 1998 season, Wise won five main events and the 250 Micro Midget Championship at Barona Speedway.

He won the USAC Championship, earning four wins at Irwindale Speedway along with rookie of the year honors.

This championship made Josh the youngest champion in USAC history, in 44 years of sanctioned races.

[1] For 2000, Wise made the move to USAC's Western States Midget Division, racing only the pavement events.

In his rookie year Wise captured two wins at Madera Speedway and one at Stockton 99, beating midget legend Sleepy Tripp to the checkers.

In 2007, Wise returned to the team to run 11 races, earning one pole position and six top 10s, including three second-place finishes.

00 Aaron's, Inc. Toyota, Wise earned two top 10s, including a career best sixth, at Las Vegas Motor Speedway.

After seven races, he caught the eye of car owner Dale Earnhardt Jr. and was hired to drive for JR Motorsports part-time.

7 JR Motorsports Chevy with sponsorship from Go Daddy and HowDoYou.com part-time through 2010 and in 2011, splitting the ride with IndyCar Series driver Danica Patrick; his average finish was 13th and he completed every race on the lead lap.

[8] Wise mostly ran the 26 as a start and park entry and finished behind Stephen Leicht in ROTY standings.

On November 26, 2013, Wise announced that he had left Front Row Motorsports;[9] on December 4 it was revealed that he would drive for Phil Parsons Racing in the 2014 NASCAR Sprint Cup Series season.

[12] By the GEICO 500, the team had struggled, and was ranked 42nd in owners points, though Wise recorded his first career top-ten in the race.

He didn't drive for Martinsville, Phoenix, and Homestead as Gray Gaulding drove for the team at those races.

[21] A user reached out to the Dogecoin community, a cryptocurrency based on the Internet meme, Doge, that had raised funds for other various causes.

Phil Parsons Racing told /r/NASCAR the communities needed to raise $55,000 to sponsor Wise at the Aaron's 499,[22] or just over 67 million Dogecoins.

Wise did not advance through the Showdown but won the fan vote, defeating favorite Danica Patrick.

[29] He currently serves as a driver performance manager for Chevrolet teams CGR, GMS Racing, and Hendrick Motorsports.

[30][31] Wise won the USAC Midget race at Lucas Oil Raceway on August 14, 2021 driving for Alex Bowman.

[33] He is a triathlete, beginning after cycling with driver Scott Speed, and occasionally tests his bicycle at the A2 Wind Tunnel in North Carolina.