You can post live videos through YouTube. Create an account, and be sure to give them access to your camera and microphone—that part’s important for streaming things.
Each live video app is a little different, but when you go into their settings you’ll look for sharing options. Turn on Tumblr sharing (log into Tumblr if prompted) and tap “Allow.” If you have more than one blog, you can use the app’s settings to pick which one you’re posting to. For more on live video, check out our Support post about it.