The "motherboard" also known as logic board is virtually the whole phone, as it contains all info about the original phone on that board. Like serial number and UD-ID, IMEI number etc, so it would practically be a totally different phone if you took it out and changed it.
But no, i haven't seen any places that sell them because of that. As it's the main part of the whole phone and would probably be expensive.
What you can do is buy one of those wifi storage devices that are small and fits in your pocket that you can buy up to 32GB and interconnects with the iPhone over wifi, though batter life in those units are just a couple of hours..
So what i'm saying is, look at buying a whole new phone as that is probably easier and cheaper in the long run.