Leonhardi Euleri Opera omnia: Descriptio commercii epistolici, Volume 1 gives detailed information where the *original* letters are, or at least were when the book was written) (I take it from your question you know of some partial printed/edited versions available at the link mentioned in a comment). Most of them are in Russia (more specifically to a large extent in the archive of the academy of science of St. Petersburg).

I do not know whether some scans are online. However it seems that the publication of the respective volume of the Opera Omnia containing the full correspondance is imminent or already happened (I found a page where it said sheduled for 2011).