Does rotating light source around the light propagation direction introduce angular momentum to the light field? If the light source is rotating around the light propagation direction $z$, will the emitted light gain extra (spin or orbit) angular momentum?
 A: For some types of the source rotating around an axis, the emitted electromagnetic wave can pick up an extra angular momentum
(rotating around the axis stated in the question is not completely clear).
First, it should be noted that spin angular momentum of light corresponds to the polarization of the emitted wave, and the (intrinsic) orbital angular momentum corresponds to the degree of twist or helicity of the wavefronts. Here I have discussed an example of the former.
Spin angular momentum:
An example of a rotating radiation source that gives rise to an spin angular momentum is a rotating dipole radiator. Consider an infinitesimal dipole located in the x-y plane at the origin:
$$\vec p = \Re\left \{ (p_x, p_y, 0)e^{-i\omega t}  \right \}$$
where $p_x$ and $p_y$ are complex numbers in general.
The electric field (phasor) of this dipole in the far-field (Fraunhoffer) region and on the z axis can easily be obtained as:
$$\vec E (r=0,z)=-\frac{\omega^2 \mu_0 }{4\pi} \frac{e^{ikz }}{z} (p_x\hat x+p_y \hat y)$$
Now, in order to see what happens for a dipole rotating around its axis (here the z-axis), we choose $p_x$ and $p_y$ such that they have $\pi/2$ phase difference and equal magnitude: $p_x=p_0$ and $p_y = p_0e^{i\pi/2}=ip_0$. This would correspond to a dipole rotating around the z-axis. The electric field would become:
$$\vec E (z)=-\frac{\omega^2 \mu_0 }{4\pi} \frac{e^{ikz }}{z}p_0 (\hat x+i \hat y)$$
which has a circular polarization and therefore carries a non-zero spin angular momentum. Note that this is caused by the mechanical rotation of the dipole and the radiated far-field of a non-rotating dipole is linearly polarized and does not carry spin angular momentum.
                                                        
This rotating dipole can be used for the simulation of spin-controlled unidirectional photonic waveguiding, which is based on spin-enabled evanescent-wave coupling of the radiated field and the guided wave. Evanescent waves are known to have an intrinsic transverse spin component coupled to the propagation direction, and therefore the propagation direction can be controlled by the spin of the wave. 
The following shows a rotating dipole coupled to a dielectric waveguide (evanescent wave coupling). The propagation direction can be controlled by rotation direction of the dipole. (taken from Rodríguez-Fortuño, Francisco J., et al. "Near-field interference for the unidirectional excitation of electromagnetic guided modes." Science 340.6130 (2013): 328-330.)

