The templates for the two constructions you are asking about look like this:
- genitive construction with apostrophe: [X]'(s) Story = The Story of [X]
- attributive construction without apostrophe: The [X] Story = The Story about/of type [X]
With a genitive apostrophe: "the" would be used only if it goes with Richmond Players
So "The Richmond Players' Story" breaks down as "[The Richmond Players]' Story" meaning "the story of [the Richmond Players]."
To tell whether this is correct, we need to know more about how the name of your group, "Richmond Players,"is used.
Some proper nouns in English are used with a definite article (sometimes thought of as being part of the name, and sometimes not), while others aren't. Baseball team names are an example of proper nouns used with a definite article: people say things like "the Houston Astros." If your name is like this, and you can be called "the Richmond Players" (e.g. in phrases like "brought to you by the Richmond Players") then "The Richmond Players' Story" would be a correct way to phrase this (even if it is not the only option).
If the group is never referred to as "the Richmond Players"—that is, if you say things like "brought to you by Richmond Players"—you would instead say "Richmond Players' Story" when using a genitive apostrophe.
Without an apostrophe: the goes with Story, and "Richmond Players" in between describes the category of the story
"The Richmond Players Story" breaks down as "The [Richmond Players] Story" meaning something like "The story about/of the type [Richmond Players]."
This structure can be used either with a proper noun that does not take a definite article, or a proper noun that does take a definite article. In the later case, you usually don't double up on the definite article. Examples:
- The Star Wars Story ("Star Wars" is a proper noun that is used without a definite article)
- The Houston Astros Story ("Houston Astros" is a proper noun that is used with a definite article, but in this construction, you only use one definite article)
I agree with the argument that you do not need to use an apostrophe: you can say "The Richmond Players Story" if you are using the construction described in this section.