Is it possible to play doom on an oscilloscope using only lissajous figures?

A custom doom renderer could produce edge lines directly. Maybe. Worst case you could render each surface a different solid color and the do edge detection / line conversion. Sprites as they point out should get separate handling.

Vector graphics are still very cool, and becoming quite rare to see in the wild.

Lissajous figures? They just reinvented the vector display. That's no more lissajous figure as a pixel-based display is Tetris (in each case you can display that, but many other things too).

Not sure I understand the lissajous figures part... isn't that an x-y plot where the x and y waveforms are both sine waves? Couldn't x and y channels be controlled more flexibly here? Or is this part of the challenge?

Now I'm wondering what I can do with my Vectrex if I replace the CPU/RAM with something modern.

Of course it's no longer a retro console, but cool to keep it in the original case with controller(s).