If, by LC oscillator, you mean a circuit composed of an 'ordinary' inductor and capacitor etc. then the answer is no.
The equations for the LC oscillator are derived within the context of ideal circuit theory which is the limit of a number of assumptions.
To apply the results from ideal circuit theory to physical systems, the physical systems must approximate the assumptions of ideal circuit theory.
One of these is that physical circuit elements can be represented with lumped element models.
And, for that to hold, the wavelengths of the signals of interest must be much larger than the dimensions of the physical circuit elements.
When this assumption does not hold, we must use the distributed element model such as in, for example, transmission line theory.
Another related consideration is that physical circuit elements have parasitic properties that cannot be avoided. Thus any physical system of conductors possess parasitic inductance and capacitance that become significant at high enough frequencies.
For visible light, the wavelengths are so small (hundreds of nanometers) that the assumptions of ideal circuit theory are not remotely valid so, unless one generalizes the notions of "inductor" and "capacitor" (and the other circuit elements) far outside the ordinary, the answer is no, an LC oscillator cannot produce visible light.