The agency won the business after a pitch that included the incumbents TDA, which handled the charity's creative work, and Mike Colling & Co, which handled media.
Rapp will now create all of the association's marketing communications, which incorporates TV, digital, press, outdoor and direct marketing work.
It will also oversee all onand offline media planning and buying.
TDA had handled the association's advertising account since May 2006. The agency's most recent campaign for the charity featured a digital push that saw the association's "Sponsor a Puppy" initiative move online.
A new website was developed to back the scheme, which was also supported by a direct mail pack that encouraged 100,000 recipients to donate £2 to the charity.