Sorry, but your Nintendo Network ID (NNID) and eShop account are locked to the 3DS that they were created on and only for you to transfer them onto another 3DS and/or New 3DS XL is by way of the "System Transfer", which does require that you still have the original 3DS in your possession and able to access to the Internet.
However, in the event of a loss such as fire or theft, Nintendo can remotely make the transfer for you, so the only way to find out is by call Nintendo's 1-800-255-3700 (6 am ~ 7 pm Pacific, everyday) Customer Service number, or use their webform to contact them, and one of the service representatives can tell you if they will allow it for upgrading to the New 3DS XL or not.
And another bit of advice, sorry, while it might be possible for them to make the transfer for you, you will lose your game saved data as that is not transferable (each 3DS encodes the data in a way that only it can read it), so you will have to restart all your eShop games after the transfer again if you do not do it by way of the above mentioned System Transfer.