I have now narrowed the steps to reproducing this issue on a newly flashed USB and boot Micro SD.
Use
this guide to enable wiimote connections.
Install the "old format hiscore.dat" in the mame2003 bios as explained in the high scores section
here.
Enable Root Login over SSH by doing the following:
sudo nano /etc/ssh/sshd_config
Change PermitRootLogin to "PermitRootLogin yes"
Enter "sudo passwd root" then enter in the root password.
Restart for changes to take effect
I'm still trying to narrow it down even farther.
So far, finishing with the Wiimote steps has triggered the issue and finishing with the hiscore.dat has triggered the issue (which is really strange).
Edit: Upon narrowing it even further. You don't need to be running off USB for this to occur. All you have to do is follow the wiimote connection guide to cause this issue.
Based on this info I can assume that it's also inconsistent when the issue will first occur after following these steps, as one reboot often wasn't enough to cause it.
I've now narrowed down the cause to the following line in "/home/pi/bin/attachwii.sh".
wminput -d -q -c /home/pi/mywminput 00:1C:BE:F2:D8:64 > /dev/null 2>&1 &
The '&' at the very end of the line is the cause of this. Removing the '&' and then running it does not cause the same issue.
Now the problem seems to be that removing the '&' at the end will not allow your system to finish booting if you have "/home/pi/bin/attachwii.sh" run on startup. Now the question is getting a little more specific as what to fix so I'll create a new post and link it below:
https://retropie.org.uk/forum/topic/7296/wminput-command-with-on-the-end-breaks-wifi