Is there a developer API?
Is there an API for developers? If not is it in the works?
Outlook Customer Manager is based on Exchange and Office365. So for now you can use existing APIs for Contacts and Tasks.
Marco Behler commented
is there a developer API to update/insert deals, customers etc.?
The is a handy app for outlook but without programmatic access to the customer and deal records it will be difficult to benefit due to multiple sources of truth.