the continuing success is the same as: the ongoing success, and is not tied to a verb tense; it functions like a regular adjective.
the continued success is the past participle of the verb continue used as an adjective.
Other examples of the past participle as adjective clarify this:
- The forgotten son means the son that was forgotten
- The broken record means the record that was broken
- The continued success means the success that was continued
Thus, one could possibly write or say: The success that was continued [by the son, for example] can be easily shown.
If you try to put the success that was continued into your sentence, it really would not work grammatically or semantically:
- [...] one which is crucial to the continued success of any organization and its customers. would mean
[...] one which is crucial to the success that was continued of any organization and its customers.
That would only apply to a past situation; not a sentence spoken or written at the present time about a situation like yours. Yours is a general statement in the present tense.