Vacuum permittivity (aka. the "electric constant") is just a way to measure the strength of the electromagnetic interaction. In the best theory that we have, quantum electrodynamics, this role would be fulfilled by the fine structure constant α, which is related to ε0 by a simple formula (i.e., they represent the same thing.) In other words, ε0 (or α) is not a property of free space, it is a property of the electromagnetic field.
To give an explanation of the permittivity of free space, it is necessary to also give an explanation of an electric field and electrical charge. Until now no attempt has been made to explain electromagnetic effects in terms of something more fundamental. I have written a technical paper (to be published in December) which attempts to show that everything in the universe, including an electric field, is obtained from the single building block of 4 dimensional spacetime. A new constant of nature is proposed which converts electromagnetic properties (including the permittivity of free space) into a distortion of spacetime (pp 13-15). Several experiments are suggested. A preprint of this paper is available at: http://onlyspacetime.com/QM-Foundation.pdf