There isn't a 'native' app for Turbine.
However, Turbine itself is designed to be responsive which means that it resizes itself automatically to work in your tablet or phone browser.
Simply enter the normal address for Turbine in your mobile browser and log in using your email address and password. The address will be something like 'yourcompany.turbinehq.com' where 'yourcompany' is your own organisation's unique Turbine account name. (Alternatively, you go to TurbineHQ.com and log in using the 'Sign In' button.)
When you have reached the Turbine dashboard, we recommend saving that link to your phone or tablet's home screen so you can get back there any time.
This guide on How-To Geek explains how to do that on Android, Apple and Windows phones and tablets: How to add websites to the home screen on any smartphone or tablet.