Is there a keyboard shortcut to show/hide the side panel?

I found that Chrome keyboard shortcuts listed "Set focus on the rightmost item in the Chrome toolbar" as F10. In my toolbar the side panel button is 3rd from the right, so I would follow with Left Left Space to open the side panel. In your case, there may be more or fewer items on the right side.

You could then use some key mapping tool, I'm not familiar with the options on all platforms, but you could map some key combination to "F10 Left Left Space" or whatever. I personally used xdotool on Linux but there are probably good options for Windows and Mac as well.

Use autohotkey to create a shortcut which only works for Chrome like this:

#If WinActive("ahk_exe chrome.exe") 
F2::Send {F10}{Left}{Left}{Space}
#If

Just to add on to Superloach's answer (which sounds like the best option for this right now), I'm on a Chromebook and to get to F10 I hold the Search button and the 10th button on the top row (not counting esc), which for me is the volume up button.