They raised £57,440 with the help of the Art Fund charity to buy the 16-minute film by acclaimed artist Tacita Dean, which shows the painter preparing for his 2016 RA show.
It is part of a collaborative project that will see exhibitions of Dean’s work at both venues and the National Gallery this year.
The film will be shown at the National Portrait Gallery as part of her portrait exhibition from March 15 to May 28 before going to the RA.
Tim Marlow, the RA’s artistic director, said the joint acquisition underlined “the truly collaborative nature of these three exhibitions”.
He said: “We are thrilled that Dean’s portrait of David Hockney will enter the RA Collection, joining a superb range of portraits of artists by fellow artists, and continuing the theme of one Academician depicting another.”
Dean, granddaughter of Ealing Studios founder Basil Dean, was nominated for the Turner Prize in 1998 for her film about sailor Donald Crowhurst who disappeared at sea in 1969 while trying to sail non-stop around the globe.
Her exhibition of still life work will run at the National Gallery at the same time as the National Portrait Gallery show and and her landscape work will be at the RA from May 19 to August 12.