You could define a global variable (outside of any function in your HTA) that contains the active application path. Your button click events can just update the value of this variable and your ApplicationInstall() sub can just read it. It would essentially act as a property of your module.
<script language="vbscript">
Dim m_strPath ' Page/Module-level variable
Sub cmdButton1_Click()
m_strPath = <application path 1>
End Sub
Sub cmdButton2_Click()
m_strPath = <application path 2>
End Sub
Sub ApplicationInstall()
If Len(m_strPath) > 0 Then
CreateObject("WScript.Shell").Run Chr(34) & m_strPath & Chr(34), 1, True
End If
End Sub
</script>