I think it is possible to approximate the result by using explicit feature mapping followed by standard PCA (which amounts to using SVD to find the eigenvectors of your covariance matrix computed from the mapped feature vectors). The idea is to map your feature vectors to a new feature space such that applying "linear" methods (your standard PCA) on the transformed features will give "equivalent" results as using the "kernel trick" to implicitly map features to a higher dimensional feature space, which is done by the Kernel PCA algorithm.
Explicit feature mapping can be done using the Nystrom method or MCMC sampling. See the docs for kernel_approximation from scikit-learn and the referenced paper therein here and the example here for more details. In particular, the MCMC sampling method is based on the method described here.
In fact, the above "trick" is not just applicable to Kernel PCA but also for other "linear" methods, e.g., linear SVM. Note that the results might differ since there is approximation involved in the feature mapping.