To me, it's not a common AmE expression.
I think it refers to the process of submitting an order, or more accurately, of
composing an order. That is, we try to give you exactly what you are asking for. (the
way your order was "made")
It's also possible they mean something like, "We try to fill requests in the order in which they were received," OR, " - as soon as we receive them," but that would be a very bad way of putting it.
Sounds to me like a really bad translation of something.
- A.
Another remote possibility, if you are ordering something to be
custom-made, "we will try to
ship your order as soon as the item has been fabricated," but "to fill" an order usually comprises the whole process, from picking/selecting/fabricating to packaging/shipping.