Here's a list of live feeds of primates at zoos, from San Diego to the UK! Thus far, I don't think anyone has a spider monkey cam :( If you discover any ape/monkey cams that I have missed, please let me know!
marmosets at the Wisconsin National Primate Center
orangutans at the Durrell Wildlife Trust in the UK
orangutans at the National Zoo in DC
orangutans and siamangs at the San Diego Zoo
Japanese macaques at the Blank Park Zoo in Iowa
Golden Lion Tamarins at the National Zoo in DC
And if you are looking for video clips, I highly recommend that you check out ARKive or National Geographic's monkey videos and ape videos!