The "General Conditions" section of the "Terms & Conditions" on the site indicates that it's the customer's obligation unless Amex decides otherwise: (quoted from the site; use of all caps is theirs)

General Conditions
The My WishList offers and special pricing are available only at mylifemycard.com/mywishlist while supplies last and cannot be accessed by any other means. Sponsor reserves the right to suspend or terminate any offer in the event the website becomes infected with a virus or is adversely affected by unauthorized intervention or participation. In the event an offer is suspended or terminated, Sponsor will provide notice thereof on the website. Sponsor is not responsible for any computer virus or damage caused to Cardmember's computer by accessing the website. Federal, state, and local taxes may apply to purchases. CARDMEMBERS ARE SOLELY RESPONSIBLE FOR APPLICABLE TAXES, INCLUDING BUT NOT LIMITED TO SALES TAX, LUXURY TAX, TITLING AND LISCENSING OF VEHICLES, FEDERAL INCOME TAXES, AND ANY AND ALL TAXES ASSOCIATED WITH A PURCHASE. At its sole discretion, American Express may provide funds to help Cardmembers defray some of the tax costs associated with this promotion. Please consult your tax advisor with any questions. The appropriate shipping and handling fees are the responsibility of the Cardmember and will be applied to all orders prior to completing purchase transaction. See each product for details on product shipping costs and delivery timeframes. If product is damaged upon receipt, Cardmember has 14 days from receipt of delivery to report the damage (except for the Panasonic Plasma TV: Cardmembers must report the damage to American Express within 48 hours of receipt). American Express is not responsible for damage to products if Cardmember fails to report the damage within the 14-day period.

Sponsor reserves the right at its sole discretion to disqualify any individual suspected of tampering with the entry process or the operation of the promotion or website, or to be acting in any manner deemed by Sponsor to be not sportsmanlike or disruptive or with intent to annoy, abuse, threaten, or harass any other person. CAUTION: ANY ATTEMPT BY A USER OR ENTRANT OR ANY OTHER INDIVIDUAL TO DELIBERATELY DAMAGE ANY WEBSITE OR UNDERMINE THE LEGITIMATE OPERATION OF THE PROMOTION IS A VIOLATION OF CRIMINAL AND CIVIL LAWS AND SHOULD SUCH AN ATTEMPT BE MADE, SPONSOR RESERVES THE RIGHT TO SEEK DAMAGES AND COSTS (INCLUDING, WITHOUT LIMITATION, ATTORNEYS' FEES) FROM ANY SUCH PERSON TO THE FULLEST EXTENT PERMITTED BY LAW