Tempo Budgets offers a REST API to programmatically integrate it to your other enterprise components. REST being based on open standards, you are free to use any web development language to access the API. Please visit JIRA's REST API Documentation to learn how to develop the client code required to call these methods.
Note that Tempo Budgets is completely based on REST but not all of its methods are intended for external usage. You should use only methods that are listed in the API documentation. Methods intended for external usage are prefixed with /api.
You can use Atlassian's REST API Browser in order to tryout the operations manually through a web interface, integrated into JIRA. The REST API Browser will allow you to see all REST methods exposed by Tempo Budgets (and other add-ons).
The REST API includes an interactive and searchable documentation accessible directly from Tempo website.