No, of course not. Intact cells have a really hard time getting into the bloodstream. If they do (say, from handling it with an open wound) the immune system might complain about being bored by such an easy challenge.

Cancers can theoretically be transmitted via transplants, but those derive from humans and are therefore much closer to the body's own cells. And even there, immunosuppressives are required to prevent rejection.

There are some cancers that originate from viruses. HPV is the most prominent, and whatever is killing the Tasmanian Tiger. There's some speculation others may as well (the red meat -> colon cancer link comes to mind). But that's not really relevant for the topic at hand.