OK, well, hiding the scrollbar on the ScrollContainer didn't work unless I disabled it. Even theme-ing it to be invisible didn't really work.
For posterity...
As per @Megalomaniak suggested, I swapped out my ScrollContainer for a Panel node then added a VSlider as a child above my VBoxContainer. My Panel was set to clip content outside of it, which cut the rest of my VBoxContainer off. I then moved the VBoxContainer until I could see the bottom of it in the Panel and noted the Y coordinate. I made this my minimum value on my VSlider.
The VSlider then emitted a value_changed signal which would updated the position of the VBoxContainer and boom! A much nicer looking way of scrolling a panel.
Thank you both so much for the advice and help!