This is a nice question. There is actually quite a bit of work which has been done along these lines, although we are a very long way from having a good understanding of how a theory of finite-type invariants should work for higher-dimensional knots.
Building on work of Habiro and Shima, Tadayuki Watanabe has pushed the idea of finite-type invariants of (ribbon) n-knots furthest, I believe, using higher-dimensional analogues of claspers. His theory is already quite impressive, and he can recover known K-theoretical calculations of characteristic classes of unknots from his formulae, and the connection with configuration space integrals is quite explicit. References:
On Kontsevich’s characteristic classes for smooth 5- and 7-dimensional homology sphere bundles math/0610292.
Configuration space integral for long n-knots, the Alexander polynomial and knot space cohomology math/0609742.
Clasper-moves among ribbon 2-knots characterizing their finite type invariants Journal of Knot Theory and Its Ramifications, 2006; 15 (9) 1163-1200
Moreover, he is building on work of Habiro and Shima.
The other people working on this, as mentioned by Dev Sinha, are Cattaneo and Rossi
(Wilson surfaces and higher dimensional knot invariants, Comm. Math. Phys. 256 (2005) 513-537)
Cattaneo, Cotta-Ramasino, Longoni (Configuration spaces and Vassiliev classes in any dimension) Alg. Geom. Topol. 2 (2002) no.39 949-1000
Configuration space integrals (including self-linking integrals as the simplest example) for 2-knots were first studied I think by R. Bott, who found a CFI invariant for 2-knots.
Configuration spaces and embedding invariants, Turkish J. Math; 20(1) (1996) 1-17.
In another direction, Greg Kuperberg has a version of the Gauss integral which works to compute the linking number of two closed submanifolds of Sn.
From the Mahler conjecture to Gauss linking forms, math/0610904.
DeTurck and Gluck have done further work in this direction. Furthermore, there is
Clayton Shonkwiler, David Shea Vela-Vick (Higher-dimensional linking integrals) math/0801.4022.
One of the basic properties of the Gauss integral is that integrand is invariant under orientation-preserving isometries of Euclidean space, which is important in geometric applications. They find a linking integral formula in higher dimensions which shares this property.