If you just want to wait for the process to exit before proceeding, you can call Process.WaitForExit().
myProcess.WaitForExit
However, you state in comments on another answer that you want to wait for specific output from the console process. If you console app/batch file or whatever is outputting things line-by-line you could read it like this:
'Your other ProcessStartInfo code here. Configure 2 extra options as below.
processStartInfo.UseShellExecute = false
processStartInfo.RedirectStandardOutput = true
Dim myProcess
Set myProcess = SystemProcess.Start(processStartInfo)
Dim stdOut
'Redirect standard output so you can read it.
Set stdOut = myProcess.StandardOutput
Do
Wait 0, 100
'Store the line in a variable and check it for whatever condition you want instead of printing as below.
print myProcess.StandardOutput.ReadLine()
Loop While (not myProcess.HasExited)
'Get any leftover output at the end of execution.
print myProcess.StandardOutput.ReadToEnd()