America doesn't have monkeys (apart from all the zoo and pet ones your law has just released onto the streets) but you should expect this type of food theft and the things thieving gulls get up to will go through the roof. Feral dogs and cats, foxes, raccoons, coyotes and so on will all get in on the act. Only, unlike the lady with the stick in the first video, no-one will be allowed to hit the animals, because that would be assault.
Farmers will take a hit to their income. The ones specialising in dairy or eggs will go bust. The ones specialising in arable will no longer be able to kill crop pests, so their yields will go down. When DDT was first introduced, American farmers hailed it as a miracle. This is a reverse of that miracle. Meanwhile, no farmer will be able to import beehives to pollinate his crops, so if the local bees can't do the job, he's doomed. Bye bye alfalfa crops.You've not quite gone back to 19th century farming techniques because you've still got machinery doing the hard labour, but farming is going to be a lot more high risk (e.g. colorado beetles can wipe out your potato crop) for poorer rewards.
The pesticide industry, pet food industry, etc go bust.
Working animals are no more. Disabled people can't have guide-dogs, hearing-dogs, assistance dogs. Customs officials and the bomb squad can't have sniffer dogs. Mountain rescue people can't have search dogs.
Law enforcement will have tons of its time wasted arresting or cautioning people for animal related crimes. That well known phrase "Why don't you go and prevent REAL crimes?" will be uttered not when folk are pulled over for speeding, but instead when folk get taken to task for accidentally standing on an ant or swatting a fly or hitting a rabbit with their car.
(Some) lawyers will make oodles of money, because of all the ridiculous lawsuits, appeals and counter-appeals which will spring up. Mrs Miggins sues the President for not removing every insect from the White House lawns before the lawnmowers cut the grass. "He willfully allowed the murder of hundreds of caterpillars and aphids!"
Also the debates will go on for YEARS about why is it legal to imprison or kill a human who kills a human and a human who kills an animal, but not to imprison/kill an animal who kills a human or kills another animal.
Oh and you can't control any disease which has an animal vector. Malaria, rabies and bovine tuberculosis are here to stay.
EDIT INSPIRED BY ANOTHER ANSWER: There will be both good and bad health implications. Removing meat, eggs and dairy takes virtually all the monosaturated fat from the human diet. This will be good for everyone's heart - no more clogged arteries. However,. it will be bad for everyone's immune system, since polyunsaturated vegetable fats are immunosuppressants, while monosaturates are immunoboosters (Reference: The Fats of Life by Prof Caroline Pond,Cambridge Uni Press. So the heart surgery part of the economy will dwindle slowly, but the manufacturers of medicines for coughs, colds and flu will take an immediate boost.