Will I get rejected because PI cannot show funding?
Actually, in the US it would be unlikely that your application would be rejected only for lack of funding. But you could well be accepted without funding if it comes to that and your background is strong enough. 

In spite of it's association with Cornel, CALS is part of the New York State university system, so NYS rules apply to many things there. 

But there may be other funding open to you than direct funding by the professor. You might be eligible for TA or RA employment. But you might also have to apply for that separately than for admission. 

But work with the professor who will know all of your options.