For example, I wonder: Does every modular lattice embed into the lattice of submodules of some module? After all, I cannot think of any relation which holds in the lattice of submodules of some module, except for modularity.
There are lots of relations satisfied in lattices of submodules besides the ones implied by modularity. For example, there is the Desarguesian identity mentioned here (which holds in any lattice of congruence relations on an algebra of a Mal'cev theory, such as the theory of modules over a ring).
So, a projective plane which is not Desarguesian gives an example of a modular lattice which is not faithfully represented in a lattice of submodules (to the collection of points and lines, adjoin a formal top and bottom to obtain a lattice). See remarks by Freyd-Scedrov in Categories, Allegories 2.156-2.157, here.
Similarly, the free modular lattice on four generators admits no faithful representation into a lattice of submodules. If you have access to the page, see Theorem 3 on page 145 of this Google book.
The lattice of submodules of a module satisfies a stronger identity, namely the Arguesian law: $$ (x_0\vee y_0)\wedge (x_1\vee y_1)\wedge (x_2\vee y_2)\leq ((z\vee x_1)\wedge x_0) \vee ((z\vee y_1)\wedge y_0). $$ You might check the short paper
Day, Alan; Jónsson, Bjarni. The structure of non-Arguesian lattices. Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 157--159.
where a “geometric” characterization of modular, non-arguesian lattices is proved.
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.
By adding a couple of conditions we can indeed obtain a representation theorem as the OP suggests. Specifically, von Neumann's coordinatization theorem says that every complemented modular lattice of order at least 4 is isomorphic to the lattice of principal right ideals of a von Neumann regular ring. See the wikipedia article on continuous geometry or any of the books on continuous geometry mentioned in this mathoverflow post.
If we replace modularity with the stronger Arguesian law mentioned in Todd and Pedro's answers, we get the more general coordinatization theorem of Jónsson, which says that every complemented Arguesian lattice with a large partial 3-frame is again isomorphic to the lattice of principal right ideals of a von Neumann regular ring. See Proposition 10.1 of this paper by Wehrung for a first order characterization of "having a large partial 3-frame".