Can I golf in my own programming language here? I ask it 'cause there is no online way to check it out.
Yes, but an interpreter or compiler is required
Languages on this site are defined by their implementation. This means that there has to be an interpreter/compiler (online or offline) in which programs can be run.
Obviously, it is most convenient if the program can be run online (such as in Try It Online), but it is perfectly acceptable if the interpreter/compiler has to be installed offline. The key is that people should be able to verify that the answer is valid.
Most Certainly Yes
Some examples of users using their own programming language: