There are so many things that can go wrong in your scenario, that you should start
simple:
(1) given a .vbs that does not try to change the registry (which Windows eagerly defends):
MsgBox "ThatsMe"
and the invocation
err := exec.Command("cmd.exe", "/c", "ThatMe.vbs").Run()
from an .exe in the same folder, I get a security alert: "Do you really want to open ThatsMe.vbs from the mapped network drive E:\". If I agree, the script is executed and the MsgBox appears. Your security settings may be so strict that you aren't even be asked.
(2) For the above invocation to work, the shell must know how to handle .VBS files. Your assoc/ftype settings may not provide this info. Then
err := exec.Command("wscript.exe", "ThatsMe.vbs").Run()
or
err := exec.Command("cscript.exe", "ThatsMe.vbs").Run()
should work - interestingly without the security warning.
(3) Instead of relying on the PATH and having/doing all files/work in the same folder, provinding full file specifications might be a good idea:
err := exec.Command(
"C:/WINDOWS/system32/wscript.exe",
"E:/trials/SoTrials/answers/10024850/go/ThatsMe.vbs").Run()
(4) If you can execute the humble ThatsMe.vbs, but your registry.vbs still fails, then you have to research who is allowed to see/change the parts of the registry you are interested in. Perhaps you have to invoke your executable as Administrator.
(5) While experimenting, I got fairly decent error messages from Go for the (un)intended nasty things I tried (using %comspec% instead of cmd.exe, bad file specs, ...). But trying to read a non-existing registry item caused a Windows Script Host error popup and no Go error. So your "absolutely nothing happens" diagnosis makes me wonder, whether Windows hides error messages from you. There are dark options like "Display a notification about every script error" in the IExplorer Advanced settings.