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
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.