Yes, you can use SublimeText to edit source code, and not just for ARM, but for anything; and many people do. It's a very popular source code editor.
The tricky part comes when you want to do more than just edit. A traditional IDE provides far more than just editing facilities, so you would have to either implement, or find someone on the internet who has already implemented, the functions you require for working with ARM. Not least of those functions are:
- Compiling the source into target binary files (usually ELF files).
- Uploading the compiled firmware into the chip
How you would do that is somewhat dependant on your host OS and what scripting tools you have available for doing the tasks. In general the compiler will be GCC, which is freely available for all the main operating systems, so if you know the correct flags to use when compiling source for your specific target then scripting the compilation shouldn't be too much of a problem.
When it comes to uploading the code, though, that all depends on how you do the uploading. Do you use a hardware programmer? Does your target chip have a bootloader installed? For either of those, what tools are available that are scripting friendly that you can use to add functionality to SublibeText?
So unless you can find a resource online where someone has actually done just what you want to do, for the same target chip or chip family as you are using, then it's going to be a lot of work to get going. You'd be better off starting with a traditional IDE and using that as a template (investigate how it compiles and uploads) and migrate it to SublimeText, or operate a hybrid environment where you do the editing in SublimeText but use the IDE to compile and upload the code.