I discovered that there were several solutions, each at various levels of features, stability, and ease of use. I tried Koding, Orion, and Cloud 9. Cloud 9 stood out as the best of the bunch with a very friendly layout, tons of supported languages, a variety of software pre-installed (Java, Python, Git, etc), and a full Linux terminal to use.
Since Cloud 9 is simply running Red Hat Enterprise Linux (RHEL), I decided to import the Dart SDK for Linux directly into a new workspace. Unfortunately, I immediately ran into a problem. I couldn't run the Dart VM because the binary had been compiled using GNU C Library (glibc) version 2.15. In fact, Google uses Ubuntu 12.04 "Precise Pangolin" to compile Dart which uses glibc 2.15. It turns out that the version of Red Hat on Cloud 9 (RHEL 6.5) only has glibc version 2.12.
I tried three different options to try and get Dart working on Cloud 9.
1) Compile Dart in a virtual machine
I downloaded Virtual Box and the latest 64-bit version of Ubuntu (13.10). Unfortunately, Virtual Box doesn't support 64-bit OSs on my machine because it apparently doesn't have some visualization enhancements. Not only that, but I also found out that Ubuntu 13.10 uses an even higher version of glibc (2.17).
2) Compile Dart directly in Cloud 9
I figured since Cloud 9 supports Java, C, Python, Git, Subversion, etc I would just try to pull in the source directly into a Cloud 9 workspace and compile it there, using this as a guide. I spent massive amounts of time trying to get this to work, but alas I was not successful. I could list all of the things I tried, but in the end it simply did not work. Whenever I would try to compile, the compiler would crash trying to compile SQLite. I believe it has something to do with the version of the compiler (Cloud 9 has GCC v4.4.7 and Dart apparently needs 4.6.0).
3) Use a patched Dart SDK
Some people were having very similar issues to me when trying to use Dart on CentOS. One guy built a patch that would pull in the dependencies directly from the Ubuntu repo and put them alongside the Dart SDK. This caused all sorts of bad things to happen such as random files getting locked by Cloud 9 itself and random seg faults.
So ... I decided to go back to step 1
1 - Again) Compile Dart in a virtual machine
I downloaded the 32-bit Ubuntu 10.04, which uses glibc 2.11, and installed it in Virtual Box. I followed this guide in order to get the latest Dart source code and build the SDK. I then zipped up the built Dart SDK, uploaded it via FTP to my server, and then pulled it back down onto my Windows machine. I extracted the SDK, and then uploaded it to Cloud 9. I did have to force all of the binaries to have executable permissions, since Cloud 9 doesn't give them it by default.
I then uploaded the default "Hello World" text reversal Dart project, and wrote simple Node.js wrappers around dart2js and pub to make them easier to access. I now have Dart compiling and running on Cloud 9!
Let me know if you have any questions or want examples/details of anything.