I would recommend simply adding these programs to your script's requirements so the script cannot be installed unless these programs are present. If GNOME is not already installed on your target system, installing those two gnome utilities will also install many many libraries from the GNOME project. As a user I would be annoyed if I had to download a few hundred megabytes of data and bloat my system just to run your tool.
Anyway, the choice is yours. As for installing, you cannot install a program globally on *nix without the root password, no. You can install locally from source with no need for passwords but I would not recommend that either.
If you know that your users will be running some kind of GNOME based, debian derivative, you could have your menu command launch
gksudo synaptic or whatever to allow the user to enter their password.
I don't know why you would want this of course. Just have an error message explaining to your user that these programs need to be installed. Why do you want them to install through your script? You are complicating your life and theirs for no real benefit.