I would like to use the class TrajectoryLinearSystem to create a time-varying LQR for a free-floating system as I do my own linearization of the trajectory (due to the quaternion state representation of the floating base). However, this class is not yet available in the current python bindings.
Is it possible to request this Python binding via StackOverflow or is it recommended to use GitHub issue tracker instead?
In this case it's totally fine to open a feature request issue on GitHub. I've confirmed that the binding is missing, so it's definitely in the realm of "feature request" and not "answering a help question".
I was going to say that this would be a great opportunity to open pull request to Drake with the new bindings code that you need (and that's still true), but in this case the inheritance hierarchy of TrajectoryLinearSystem is somewhat complicated so the bindings will be slightly more difficult than usual to write.