The answer is June 11, 1998.
The year-by-year timeline in Table 3 on page 13 of the attached PDF
document is the authoritative source of that information:
"11th June 1998 - Music store opens at Amazon.com"
Centre for Advanced Spatial Analysis: "Finding the Source of the
Amazon.com - Examining the Hype of the Earth's Biggest Bookstore," by
(The above is a PDF document that requires Adobe Reader to access. In
the unlikely event that this tool is not installed on your computer,
you can conveniently download it at no charge from this site):
The following Google search uncovered the information:
amazon.com bezos timeline music
I am confident that this is exactly what you are looking for, and I
happy to be able to get it to you promptly. If anything is unclear,
please ask for clarification before rating the answer.