Do AC current sources exist and why or why not? For both actual and theoretical
Ideal AC and DC voltage and current sources are useful for analysis (theoretical) purposes.
It could be said that an AC generator that supplies power to a grid operates as an AC current source. There are also AC motor controllers that act as current sources. However it may be more accurate to say that they are DC current sources that control motors through inverters. That is a DC current source drives an inverter that supplies current to a motor. Grid connected solar and wind power inverters are similar.
Re edit:

Like can I do the same thing in Chegg where you put a wave symbol and an arrow and call it a theoretical AC current source? How would you then use it in computations? Do you just attach a phasor to it and use current in the x-axis instead of voltage for vector diagrams?

Yes
Re Comment:
Here is an example from Scott, Linear Circuits, 1960. It clearly shows an AC current source, but the symbol doesn't include as sin wave symbol. I also found textbook problems that simply identified the source i(t). Some of those were sine wave sources and others defined some other periodic waveform. 
Also note that a current transformer act as a current source.