御注文頂いた商品は、本日うさぎ便（お問い合わせ番号123-4567）にて発送致しました。

Excite翻訳: The ordered commodity was sent out by the rabbit mail (tracking number 123-4567) today.
@nifty翻訳: I shipped the ordered goods in rabbit facilities (inquiry number 123-4567) today.
Babel Fish: This day rabbit flight (inquiry number 123-4567) being similar, it dispatched the order.
Google翻訳: Order your item is capped, rabbit flights today (contact number 123-4567) has been at the ships.
Yahoo!翻訳: The article which had you order it sent it out by a rabbit service (inquiry number 123-4567) today.
OCN翻訳: I shipped off the goods you ordered in rabbit flight (question number 123-4567) today.
Windows Live Translator: Your order snow-capped commodity today lepus stool (contact number 123 - 4567) loading dock.

Summary:
"お問い合わせ番号" is used for telephone number, too.
But, surprisingly, Excite somehow knows it is for shipment.
Excite's is good in general, and OCN's sounds good, too, as far as its meaning is focused on.