Is there an online bash script testing site?

I believe you are looking for a "free shell account". You should be able to sign up for one that has a description of what you are looking for.

These are typically provided with no expectation of privacy. Be careful what you store and what passwords you use.

The #bash channel on freenode recommends ShellCheck, an online resource where you can paste your shell scripts and they will be checked for syntax errors, POSIX compliance and other straightforward errors:

ShellCheck is a static analysis and linting tool for sh/bash scripts. It's mainly focused on handling typical beginner and intermediate level syntax errors and pitfalls where the shell just gives a cryptic error message or strange behavior, but it also reports on a few more advanced issues where corner cases can cause delayed failures.

ShellCheck is an open source project, so you can download the code and run your own instance.

For a heavy weight solution, you can spin up an Amazon EC2 instance. The "free/trial" tier is free for the first year of your account:

http://aws.amazon.com/free/

So, in that case, you'll have a wide choice of distros and shells to play with.