NB As mentioned by Richard Deeming below, this is only valid with .NET Core.
Most of those extra quote characters are unnecessary. Try this:
Dim myproc As New Process
Dim start As New myproc.StartInfo
start.FileName = "C:\Windows\SysWOW64\wscript.exe"
start.ArgumentList.Add("C:\Temp\AnyProcess.vbs")
start.ArgumentList.Add("Cmd1Par1=AnyVal")
start.ArgumentList.Add("Cmd1Par2=AnyVal2")
start.ArgumentList.Add("Cmd1Par3=AnyVal3")
start.ArgumentList.Add("Cmd1Par4=AnyVal4")
start.ArgumentList.Add("Cmd1Par1")
start.ArgumentList.Add("Cmd1Par2")
start.ArgumentList.Add("Cmd1Par3")
start.ArgumentList.Add("Cmd1Par4")
myproc.Start()