@JordanBklyn
First you need to confirm what version of the Nexus 7 you have. There were two devices, one with a Tegra3 SOC (2012 model) and another one with Qualcomm SOC (2013 model). Most likely the 2013 model also has an eMMC, but I can't confirm that at the moment.
If it's a 2012 model, you don't need to worry about bad blocks or anything.
You can just connect the eMMC to a PC via an USB SD card reader.
This is a board I made some time ago for the Kindle 3W:
It uses a 32GB eMMC and Microchip USB2240 - multi format flash media controller.
It can also be connected directly to a PC.
Under Linux:
"usb 3-1: new high-speed USB device number 52 using xhci_hcd
usb 3-1: New USB device found, idVendor=0424, idProduct=2240
usb 3-1: New USB device strings: Mfr=1, Product=2, SerialNumber=3
usb 3-1: Product: Ultra Fast Media
usb 3-1: Manufacturer: Generic
usb 3-1: SerialNumber: 000000225001
usb-storage 3-1:1.0: USB Mass Storage device detected
scsi20 : usb-storage 3-1:1.0
scsi 20:0:0:0: Direct-Access Generic Ultra HS-COMBO 1.98 PQ: 0 ANSI: 0
sd 20:0:0:0: Attached scsi generic sg3 type 0
sd 20:0:0:0: [sdd] 61997056 512-byte logical blocks: (31.7 GB/29.5 GiB)
sd 20:0:0:0: [sdd] Write Protect is off
sd 20:0:0:0: [sdd] Mode Sense: 23 00 00 00
sd 20:0:0:0: [sdd] No Caching mode page found
sd 20:0:0:0: [sdd] Assuming drive cache: write through"
Under Windows it's also mounted as a regular SD card.
It sounds easy because it is, you just need to power the eMMC properly. Usually they have two power rails, one for the controller/nand (VCC) and one for the I/Os (VCCQ). If you are connecting this to a 3.3V controller, be sure that you connect VCCQ to 3.3V as well.
Also, use really thin wires if you solder to that BGA, those pads can come off really easily.
@mikeselectricstuff
I usually use eMMCs/eNANDs/eSDs with SOCs via SDIO/SDMMC interface, and they can work in x1/x4/x8 mode. Regarding off the shelf readers, I tried with two chipsets, from microchip, it worked in x1 or x4 properly. But yes, maybe there are card readers out there that can work only in x4 mode, in that case another 3 wires need to be soldered.