Police were called to Poplar, just after 5pm on May 7 to reports of a dog attack, where they found a woman with a leg injury which did not require hospital treatment.
The dogs’ owner was tasered to the floor by officers and the dogs shot dead during the incident.
The petition, backed by animal rights campaigners, describes the shooting of the dogs, both as “barbaric”.
It comes after activists with the Animal Rising group gathered outside New Scotland Yard on Westminster Embankment on 11 May in protest.
The police watchdog has said it is investigating the incident.
The Independent Office for Police Conduct (IOPC) said it is “appropriate” that complaints linked to the incident on May 7 should be “independently investigated”.
Amanda Rowe, an IOPC regional director, said: “We understand the public concern regarding this incident and it is appropriate that it should be independently investigated.
“We will examine whether the actions of the officers involved were reasonable and proportionate in all of the circumstances and in line with relevant policy and procedure.”
The Met has defended its actions, saying police officers “have a duty to act where necessary”.
The dog’s owner was charged with being the person or owner of a dog that was dangerously out of control and being in possession or custody of a dog whilst disqualified from owning or keeping a dog.
He is set to appear at Snaresbrook Crown Court next month.