Unlike most other Great Ones, Oedon is only mentioned and is not seen at any time during the game. This could be because he has managed to ascend beyond even the other Great Ones, appearing only as a presence or maybe a formless voice.
Oedon hails from an ancient land, similar to the other Great Ones. While the other Great Ones learned to ascend to a higher plane than any other race that would come after them, Oedon somehow managed to ascend to a higher plane than the rest. This resulted in him never having to have a physical form. This is evidenced by the rune shown above, titled Formless Oedon.
Both Oedon, and his inadvertent worshippers, surreptitiously seek the precious blood. 
- Oedon might have used Arianna's position as a whore and her origin as a distant relative to Queen Annalise to use the blood she was exchanging with people to impregnate her. The latter seems to be more correct since the Third Umbilical Cord collected from her child states: "Every Great One loses its child, and then yearns for a surrogate, and Oedon, the formless Great One, is no different. To think, it was corrupted blood that began this eldritch liaison."
- Multiple citizens of Yharnam possess one of Oedon's two runes. This could mean that they either directly or indirectly worship him.