If by "coordinate representation" you mean the assignment of geometric positions to a graph that was initially given as a purely combinatorial structure, then one possible contender is the proof of Steinitz's theorem by
Steinitz, E. (1922), "Polyeder und Raumeinteilungen", Encyclopädie der mathematischen Wissenschaften, Band 3 (Geometries), pp. 1–139.
Somewhat later we have the proof of Fáry's theorem by
Wagner, Klaus (1936), "Bemerkungen zum Vierfarbenproblem", Jahresbericht der Deutschen Mathematiker-Vereinigung 46: 26–32.
(Fáry's and Stein's independent discoveries were later). Another would be and the drawings of sociograms in
Moreno, J. L. (1934), Who Shall Survive?, New York, N.Y.: Beacon House.