If you are able to checkout the VPN should be working fine. There are a lot of details missing, for example the protocol, and in case of http/https if you have a proxy server between your computer and the SVN server. But it seems unlikely that you would get an Access denied error from network issues.
I would look at would be the server configuration. Are you able to checkout/commit if you are NOT using the vpn (ie, locally on the SVN server or from a computer nearby the server)? Do you have authorization rules in the repository that disable write access for your user?