I think I need to use the Poynting vector to do this, as it is equal to the power per unit area flowing through a surface.
At a distance $d$km from the transmitter, the surface will have a surface area of $A = 4\pi d^2$, giving a Poynting vector of $N = \frac{P}{4\pi d^2}$.
However I am not sure how to use this to find the electric field strength.
You are already on the right track by finding $$N=\frac{P}{4\pi d^2}.\tag{1}$$
You have the definition of the Poynting vector $$\mathbf{N}=\frac{1}{\mu_0}\mathbf{E}\times\mathbf{B},$$ or taking the amplitudes only (give or take a factor of $2$): $$N=\frac{1}{\mu_0}EB.\tag{2}$$
You also know that in an electromagnetic wave the field strength amplitudes are related by $$E=cB.\tag{3}$$
From (1), (2) and (3) you can eliminate $N$ and $B$, and then find $E$ in terms of $P$ and $d$ only.
