In short, supply management is a new way of operating, （involving internal operations and external suppliers）to achieve advances in cost management, product development, cycle times, and total quality control.

我在句中加了个括号，以帮助你理解句子结构。如果我们把括号部分去掉，全句为：In short, supply management is a new way of operating to achieve advances in cost management, product development, cycle times, and total quality control.  这里应该是个a new way to do sth 的句子结构。因此，括号部分我理解为现在分词作非限定性定语。

involve sb to do 有这样的用法，特别是口语中有人这样用，但不如involve sb (in) doing sth更为规范。