Is there a way one can get Yes Private card? I know HDFC Infinia used to be an invite only but people now get it pretty easily.
Is there a way I can apply for it C2C or income based?
No, c2c is not possible for Yes Private. You can request them for Yes Private, but you need to hold the Yes First Exclusive first, then you can put up the request. And they'll only grant if you're having 45L+ of Net Income shown in your ITR