Algorithms for rank-1 updates of the SVD (also called incremental SVD) do exist, but I haven't been able to find a LAPACK-like implementation anywhere.
The one I've seen mentioned repeatedly is that of Brand (2003). Judging from this website, it seems as though Brand's algorithm is relatively simple to implement using existing LAPACK and BLAS routines as building blocks, which could perhaps explain why none of the big names has bothered to write a specialized implementation.
You can find a MATLAB implementations of various algorithms here, here, and here. The last link, to IncPACK, also mentions a C++ implementation buried within a development branch of Trilinos.
A Python implementation of Brand's algorithm (from a 2006 paper of his) can be found buried in the
svdAddCols method of the
LsiModel class of the
gensim package, version 0.5.0. (Note: I could not find this method in
gensim version 0.8.4; a similar method,
svdUpdate is present in
gensim version 0.7.4.)
A different algorithm, by Gu, is implemented in the package isvd, which is in C or C++ (haven't looked at the source to tell which; I arrived at this project from the link here, which discusses compilation flags for isvd). Another C++ (unfortunately, Windows-based) implementation is buried within a Netflix recommendation algorithm here; Netflix recommendation algorithms seem to use incremental SVD frequently, and might be another potential source of implementations.
That's the best I could do with my Google-fu.