This is a very rich and active subject. There are lots of different approaches to the problem, giving more or less strong results -- you can try to interpolate any or all of { Hecke eigenvalues, Fourier coefficients, L-values, Galois representations }, for forms satisfying various different flavours of finite-slope condition, while the weight varies in families having different numbers of parameters. Here are a selection of the important works on this:

 - For ordinary families appearing in Betti cohomology, see [Tilouine--Urban, "Several-variable $p$-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations"][1], Ann Sci ENS, 1999.
 - For non-ordinary families appearing in functions on a compact form of $Sp(4)$, try the [PhD thesis of Daniel Snaith][2] (aka "Caribou"), (Imperial College, early 2000's). 
 - For a coherent-cohomology approach, more in the style of Coleman's work for $GL(2)$, try [Andrei Jorza's thesis][3] (Princeton, 2010), or for rather stronger results [Andreatta--Iovita--Pilloni][4] (Annals, this year). 
 
These are just the references I know that treat Siegel modular forms specifically; there are other references that treat general reductive groups from which one can extract something for $Sp(4)$.


  [1]: http://eudml.org/doc/82495
  [2]: http://wwwf.imperial.ac.uk/~buzzard/maths/research/notes/sp2n.pdf
  [3]: http://www3.nd.edu/~ajorza/papers/thesis-20100428.pdf
  [4]: http://annals.math.princeton.edu/2015/181-2/p05