"吃广告饭的人" can be translated into English as "people who make a living in the advertising industry" or more concisely as "advertising professionals" (though the latter is a broader term that may include roles beyond just those directly earning from advertising work, it's often understood in context).
For a more literal and specific translation that captures the essence of "earning one's living through advertising," "people who earn their bread and butter from advertising" could also be used, though it's a bit more idiomatic and less commonly used in professional contexts.
In most business or professional discussions, "people in the advertising business" or "advertising practitioners" would be appropriate and easily understood.