The Third Doctor constructed the Whomobile shortly after Jo's departure. (AUDIO: Terror of the Master) According to the Brigadier, the Doctor had used UNIT funds to build both Bessie and the Whomobile, which he considered a potential misuse of public funds. (PROSE: Hello Goodbye) The Doctor also claimed that his "new car" was "carbon negative", as it absorbed carbon dioxide rather than emitting it. (AUDIO: Terror of the Master)
At some point, the Whomobile was put in mothballs. It was taken out when the Remoraxians took over a UNIT seabase. The Doctor and Sarah Jane took the Whomobile to quickly get to the base. (COMIC: In With the Tide)
Lupton stole the Whomobile after taking the Doctor's Metebelis crystal at the behest of the Eight Legs. After Lupton abandoned it and stole a gyrocopter, the Doctor and Sarah Jane used the craft's flying ability to follow him. (TV: Planet of the Spiders)
Slist Fayflut Marteveerthon Slitheen, a Slitheen, impersonated the Ninth Doctor, starring in the Holovid show Doctor Who?. They drove their own version of the Whomobile, and used it to save the real Doctor from a group of fake Chumblies. (COMIC: Doctormania)
Behind the scenes
The vehicle was not created for Doctor Who, but was personally commissioned by Jon Pertwee, who retained possession of it after his tenure as the Doctor. Pertwee gave it the name "Whomobile" during press interviews, but it was also known — again, off-screen — as "Alien". Had the vehicle been named in Doctor Who, it almost certainly would not have been called "the Whomobile", as this violated producer Barry Letts's strong conviction against making plays on the programme's title. The Doctor Who Technical Manual referred to it simply as "the Doctor's Car." In spite of this, the name was indeed given as "the Whomobile" in the audio drama featuring Pertwee and Elisabeth Sladen, Glorious Goodwood.
As a real-life vehicle
The Whomobile (reg: WVO 2M) was one of Jon Pertwee's personal vehicles. He retained possession of it until nearly the end of his life.
The Whomobile was hastily written into the script of Invasion of the Dinosaurs, where it replaced an army motorcycle as the Doctor's transport around the deserted London. The Whomobile's roof/door section had not been completed at the time of filming, so a motor-boat windscreen was fitted to make the car legally roadworthy. The vehicle was classified technically as an "Invalid Tricycle".
When Pertwee demonstrated the vehicle to Peter Purves on the 5 November 1973 edition of Blue Peter during his tenure, he revealed that the vehicle was roadworthy and legal, though he mentioned that the BBC forbade him or any of the production crew from removing it from the studios except for outside shoots. It was capable of a top speed of 105 mph (roughly 169 km/h). Footage of this is available on the DVD release of Invasion of the Dinosaurs and the Season 10 Blu-ray.
An extra on The Three Doctors DVD reveals Jon Pertwee introduced it on Blue Peter prior to its debut.
As documentary subject
Jon Pertwee presented the Whomobile in the documentary 30 Years in the TARDIS.
Jon Pertwee, in costume and in character as the Doctor, appeared with the Whomobile in Billy Smart's Children's Circus on the afternoon of Sunday 4 November 1973 (broadcast Sunday 6 January 1974). The Doctor Who theme music played as the Doctor drove the vehicle into the circus ring, with Gabriella Smart, sister of young ringmaster David Smart Jr., in the passenger seat. The Doctor then chatted briefly with David, before driving off with Gabriella to help find her lost performing dogs.