Yes, you can use a local environment to develop. For example, this answer contains a complete start to authenticating an HTML+javascript app. All it requires is for you to run XAMPP (or similar server) on your local machine.
If you don't need authorization, you can even host your app on one of the free services. For example, this single-web-page app hosted on jsBin.
Other platforms (Python, Java, etc.) should be just as amenable.
One caveat is that the Stack Exchange authentication methods currently don't seem to handle non-standard ports very well (there are some bug reports about this).
Also, for local development, you normally will want to set your app with:
- "Client Side Flow" enabled
- "Desktop OAuth Redirect Uri" enabled