It is quite easy to compute the first derivative of vapor pressure with respect to temperature from a cubic equation of state at least at the critical point since there is a continuity with the critical isochore. But, is there any way to establish the second derivative or at least its value at the critical point? For simplicity, Van der Waals or Redlich-Kwong, Soave-Redlich-Kwong or Peng-Robinson equations of state are in my mind.
I believe the answer to your question can be found in this article : "On the continuity of isochore slopes, and the divergence of the curvature of the vaporization curve at the critical point of a simple fluid" by John Stephenson.
From what I understand, it can be shown that the second derivative of saturation pressure is proportional to the isochoric heat capacity of vaporization. Because of the divergence of this quantity at the critical temperature, the second derivative of saturation pressure will, theoretically, also go to infinity.
Anyway, cubic equations of state do not reproduce the divergence of isochoric heat capacity so you might obtain a finite value if you apply the equations from Stephenson's article and combine it with the isochoric heat capacity of vaporization from the cubic equation (which is also related to the second derivative of the alpha function in these equations).