Is there a way that I can get a version of the Stack Overflow application to deploy it at our organization and use it?
(an official localized Stack Overflow application)
No, the software is not available for use outside of Stack Exchange.
Please see: https://meta.stackexchange.com/a/88753
Stack Exchange no longer offers a subscription service nor licenses the software to host your own Stack Exchange sites. Stack Exchange sites can only be created through the democratic, community-driven process at Area 51.