I don't think I even know a natural non-zero map from the space $S_2(\Gamma_1(N);\mathbf{Q}_p)$ ("classical" modular forms with $p$-adic coefficients, defined for example as global sections of an appropriate sheaf on the modular curve $X_1(N)/\mathbf{Q}_p$) to the space $H^1(X_1(N),\mathbf{Q}_p)$ (etale or singular cohomology).
The problem is that the usual Eichler-Shimura isomorphism works when the coefficient fields are both the complex numbers, but the definition of the map itself involves an integral and is "transcendental" in nature. The two spaces above are, loosely speaking, $\mathbf{Q}_p$-subspaces of some $\mathbf{C}$-vector space, but they're not the same $\mathbf{Q}_p$-subspace -- the difference between them is some period, which is related to some $L$-value, which is probably not an algebraic number in general.
On the other hand, the systems of Hecke eigenvalues showing up in both spaces are the same, because they are isomorphic once you base extend to the complexes.
But if one can't write down a map between the finite-dimensional spaces, is there any hope of doing it with these fancy infinite-dimensional generalisations? This isn't a proof that one can or can't do it, but perhaps it's some food for thought.
So then you have to start thinking about this work of Iovita, as David already mentioned, and things get much more complicated quite quickly.
Coleman also wrote a paper called something like "a $p$-adic Eichler Shimura isomorphism" -- I forget the precise title -- but again you might find that it's not really what you're looking for.