From Homestar Runner Wiki
The Puke Pail is a green bucket covered in puke stains designed for puking into. A "Puke Pail for Him!" is owned by The Brothers Strong and appears in the not-quite-Strong Bad Email short Sick Day. A sick Strong Bad orders The Cheat to bring him the puke pail and some more tissues twice. However, The Cheat was inside the puke pail, making crying noises and, presumably, puking or in between fits of puking. Coach Z brings a "Puke Pail For Whoev!" while talking to people, as people frequently puke during his conversations.