It's not accurate to say that no theory of integration on infinite-dimensional spaces exists. The Euclidean-signature Feynman measure has been constructed -- as a measure on a space of distributions -- in a number of non-trivial cases, mainly by the Constructive QFT school in the 70s.
The mathematical constructions reflect the physical ideas of effective quantum field theory: One obtains the measure on the space of field histories as the limit of a sequence/net of "regularized" integrals, which encode how the effective "long distance" degrees of freedom interact with each other after one averages out the short distance degrees of freedom in various ways. (You can imagine here that long/short distance refers to some wavelet basis, and that we get the sequence of regularized integrals by varying the way we divide the wavelet basis into short distance and long distance components.)
I don't think the main problem in the subject is that we need some new notion of integration. The Feynman measures we mathematicians can construct exhibit all the richness of the "higher categories" axioms, and moreover, the numerical computations in lattice gauge theory and in statistical physics indicates that the existing framework is at the least a very good approximation.
The problem, rather, is that we need a better way of constructing examples. At the moment, you have to guess which family of regularized integrals you ought to study when you try to construct any particular example. (In Glimm & Jaffe's book, for example, they simply replace the interaction Lagrangian with the corresponding "normally ordered" Lagrangian. In lattice gauge theory, they use short-distance continuum perturbation theory to figure out what the lattice action should be.)
Then -- and this is the really hard and physically interesting part -- you have to have enough analytic control on the family to say which observables (functions on the space of distributions) are integrable with respect to the limiting continuum measure. This is where you earn the million dollars, so to speak.