I also think it's likely that you will have trouble finding a replacement logic board.
As far as saving data that wasn't backed up, all of the 2013 MacBook Pros used flash storage. The Early 2013 models appear to have used SATA SSDs. The Late 2013 models appear to have used PCIe ones. Other World Computing sells replacement SSDs and external transfer enclosures for the Late 2013 models.
So it sounds like it should be possible to remove the SATA SSD or PCIe circuit board SSD from your 2013 MBP, place it into an external enclosure, and attempt to recover the contents using another Mac. This might be a way to proceed if you cannot find a replacement logic board, or the cost of replacing the logic board would be too high, relative the cost of replacing the computer. Maybe the data on the drive can be saved, maybe not – but if you can remove it and put it into an enclosure, at least you can try.
Note that on some computers with flash storage, the SSDs / flash chips are soldered directly to the logic board, and thus replacing the logic board wouldn't save your data. I believe that current Mac laptops are like that, underscoring the importance of keeping proper backups.