If this is a sign that will be posted in a store, then yes, it's correct. It's rather long, though. It would look best with the explanation ("In an effort...") in a small font, and the final line ("drinking straws served only upon request") in a larger font, centered, below the other text.
If this is not a sign or poster, then you need the verb "are" after the words "drinking straws" ("drinking straws ARE served only upon request"). Another possibility is the verb "will be," or the verb "were" (if the sentence is in past tense).