It's literal: it means "using oil to kill puppies."
Imagine that you are the owner of a large company that makes oil. Your company kills puppies, because it makes the oil cheaper. Public Relations is the name of the part of your company which tries to make people forget that your company kills puppies.
For example, your company has a big tank of oil, and the tank breaks, and all the oil comes out, and some puppies drown because they can't swim in oil. Your public relations department will try to make sure that people do not get angry about that.
It is a humorous exaggeration of what large oil companies actually do.
"With oil" is not a complete sentence - it is the end of the sentence:
They’re the people who try to make sure everyone sees the corporation as the good guys. Even when they kill puppies. With oil.
which uses full stops like "big commas" for dramatic effect. When spoken, the pause makes the listener think the sentence has finished, but then you tell them more.