Faigle and Herrmann, "Projective Geometry on Partially Ordered Sets," Transactions of the American Mathematical Society 266 (1981), 319-332. (There is an error in the statement of Corollary 4.8.)
http://www.ams.org/journals/tran/1981-266-01/S0002-9947-1981-0613799-9/S0002-9947-1981-0613799-9.pdf

There are also articles, a book and a thesis by Stefan E. Schmidt on topics like "projective geometry on an ordered set of points."

Edit: Schmidt, Stefan E. Grundlegungen zu einer allgemeinen affinen Geometrie. Birkhäuser Verlag, Basel, 1995.

Schmidt, Stefan E. Projektive Räume mit geordneter Punktmenge. Mitteilungen aus dem Mathematischen Seminar Giessen, No. 182 (1987).

There is also Benson, D. J. and Conway, J. H., "Diagrams for modular lattices,"
J. Pure Appl. Algebra 37 (1985), no. 2, 111–116, although its representation is not as good as Faigle and Herrmann's. I just list it since it's Conway.

I haven't read J. Yves Semegni's thesis, "ON THE COMPUTATION OF FREELY GENERATED MODULAR LATTICES," but it discusses a representation in Section 5.4.
https://scholar.sun.ac.za/handle/10019.1/1207

Then there is Marcel Wild's unpublished manuscript, "Modular Lattices of Finite Length." It is available on his website.

Further edit: These references give representation theorems for modular lattices, at least those of finite height. They do not give a representation theorem in terms of lattices of submodules, for the reasons others have stated, but they do give representations in terms of something like a projective geometry on a partially ordered set of points.