Their own website makes no reference to an official API, and search engine results don't point to one either.
Unfortunately, some applications, especially web apps, are not designed for accessibility/automation.
Depending on how the interface of the application re-arranges itself, you could try using location-based commands to move the cursor, E.G. "click the third channel", rather than a specific name of a channel which may be located elsewhere.
If the application has keyboard support, that may be even better, doing away with the variability of interface element locations, especially if you're switching between fullscreen and windowed modes (or screens with different resolutions/aspect ratios/scaling), though the basic principle of using numbered locations would still need to remain.
Learning how to reverse-engineer an application is unlikely to be a recommended introduction to programming in general, and is most certainly outside the scope of VoiceAttack itself.