The problem appears to be that linux-headers-4.9.67-v7 are not present.

Starting successfully at step #4, and proceeding OK to step #8, item #6 command failed:
 dkms build -m ${DRV_NAME} -v ${DRV_VERSION}

pi@raspberrypi:~ $ sudo apt install linux-headers-$(uname -r) build-essential dkms gitReading package lists... Done
Building dependency tree       
Reading state information... Done
E: Unable to locate package linux-headers-4.9.67-v7
E: Couldn't find any package by glob 'linux-headers-4.9.67-v7'
E: Couldn't find any package by regex 'linux-headers-4.9.67-v7'
pi@raspberrypi:~ $ sudo apt install linux-headers-$(uname -r) build-essential dkms gitReading package lists... Done
Building dependency tree       
Reading state information... Done
E: Unable to locate package linux-headers-4.9.67-v7
E: Couldn't find any package by glob 'linux-headers-4.9.67-v7'
E: Couldn't find any package by regex 'linux-headers-4.9.67-v7'
pi@raspberrypi:~ $ git clone https://github.com/lkw16/rtl8814AU.git/
Cloning into 'rtl8814AU'...
remote: Counting objects: 594, done.
remote: Total 594 (delta 0), reused 0 (delta 0), pack-reused 594
Receiving objects: 100% (594/594), 2.14 MiB | 1.16 MiB/s, done.
Resolving deltas: 100% (235/235), done.
pi@raspberrypi:~ $ cd rtl8814AU
pi@raspberrypi:~/rtl8814AU $ sudo su
root@raspberrypi:/home/pi/rtl8814AU# DRV_NAME=rtl8814AU
root@raspberrypi:/home/pi/rtl8814AU# DRV_VERSION=4.3.21
root@raspberrypi:/home/pi/rtl8814AU# mkdir /usr/src/${DRV_NAME}-${DRV_VERSION}
root@raspberrypi:/home/pi/rtl8814AU# git archive driver-${DRV_VERSION} | tar -x -C /usr/src/${DRV_NAME}-${DRV_VERSION}
root@raspberrypi:/home/pi/rtl8814AU# dkms add -m ${DRV_NAME} -v ${DRV_VERSION}
bash: dkms: command not found
root@raspberrypi:/home/pi/rtl8814AU# sudo apt-get install dkms
Reading package lists... Done
Building dependency tree       
Reading state information... Done
The following additional packages will be installed:
  cpp-4.6 gcc-4.6 linux-headers-3.6-trunk-common linux-headers-3.6-trunk-rpi linux-kbuild-3.6
Suggested packages:
  gcc-4.6-locales python3-apport menu libmudflap0-4.6-dev gcc-4.6-doc libgcc1-dbg libgomp1-dbg
  libquadmath-dbg libmudflap0-dbg libcloog-ppl1 | libcloog-ppl0 libppl-c4 libppl12 | libppl9
  binutils-gold
The following NEW packages will be installed:
  cpp-4.6 dkms gcc-4.6 linux-headers-3.6-trunk-common linux-headers-3.6-trunk-rpi
  linux-kbuild-3.6
0 upgraded, 6 newly installed, 0 to remove and 0 not upgraded.
Need to get 13.3 MB of archives.
After this operation, 49.6 MB of additional disk space will be used.
Do you want to continue? [Y/n] y
Get:1 http://raspbian-us.ngc292.space/raspbian stretch/main armhf cpp-4.6 armhf 4.6.4-5+rpi1 [4,145 kB]
Get:2 http://raspbian-us.ngc292.space/raspbian stretch/main armhf dkms all 2.3-2 [74.8 kB]
Get:3 http://raspbian-us.ngc292.space/raspbian stretch/main armhf gcc-4.6 armhf 4.6.4-5+rpi1 [4,363 kB]
Get:4 http://raspbian-us.ngc292.space/raspbian stretch/main armhf linux-headers-3.6-trunk-common armhf 3.6.9-1~experimental.1+rpi7 [4,189 kB]
Get:5 http://raspbian-us.ngc292.space/raspbian stretch/main armhf linux-kbuild-3.6 armhf 3.6-1~experimental.1+rpi2+b2 [136 kB]
Get:6 http://raspbian-us.ngc292.space/raspbian stretch/main armhf linux-headers-3.6-trunk-rpi armhf 3.6.9-1~experimental.1+rpi7 [414 kB]
Fetched 13.3 MB in 4s (2,800 kB/s)                   
Selecting previously unselected package cpp-4.6.
(Reading database ... 123050 files and directories currently installed.)
Preparing to unpack .../0-cpp-4.6_4.6.4-5+rpi1_armhf.deb ...
Unpacking cpp-4.6 (4.6.4-5+rpi1) ...
Selecting previously unselected package dkms.
Preparing to unpack .../1-dkms_2.3-2_all.deb ...
Unpacking dkms (2.3-2) ...
Selecting previously unselected package gcc-4.6.
Preparing to unpack .../2-gcc-4.6_4.6.4-5+rpi1_armhf.deb ...
Unpacking gcc-4.6 (4.6.4-5+rpi1) ...
Selecting previously unselected package linux-headers-3.6-trunk-common.
Preparing to unpack .../3-linux-headers-3.6-trunk-common_3.6.9-1~experimental.1+rpi7_armhf.deb ...
Unpacking linux-headers-3.6-trunk-common (3.6.9-1~experimental.1+rpi7) ...
Selecting previously unselected package linux-kbuild-3.6.
Preparing to unpack .../4-linux-kbuild-3.6_3.6-1~experimental.1+rpi2+b2_armhf.deb ...
Unpacking linux-kbuild-3.6 (3.6-1~experimental.1+rpi2+b2) ...
Selecting previously unselected package linux-headers-3.6-trunk-rpi.
Preparing to unpack .../5-linux-headers-3.6-trunk-rpi_3.6.9-1~experimental.1+rpi7_armhf.deb ...
Unpacking linux-headers-3.6-trunk-rpi (3.6.9-1~experimental.1+rpi7) ...
Setting up cpp-4.6 (4.6.4-5+rpi1) ...
Setting up linux-headers-3.6-trunk-common (3.6.9-1~experimental.1+rpi7) ...
Setting up dkms (2.3-2) ...
Setting up gcc-4.6 (4.6.4-5+rpi1) ...
Processing triggers for man-db (2.7.6.1-2) ...
Setting up linux-kbuild-3.6 (3.6-1~experimental.1+rpi2+b2) ...
Setting up linux-headers-3.6-trunk-rpi (3.6.9-1~experimental.1+rpi7) ...
Examining /etc/kernel/header_postinst.d.
run-parts: executing /etc/kernel/header_postinst.d/dkms 3.6-trunk-rpi
root@raspberrypi:/home/pi/rtl8814AU# dkms add -m ${DRV_NAME} -v ${DRV_VERSION}

Creating symlink /var/lib/dkms/rtl8814AU/4.3.21/source ->
                 /usr/src/rtl8814AU-4.3.21

DKMS: add completed.
root@raspberrypi:/home/pi/rtl8814AU# dkms build -m ${DRV_NAME} -v ${DRV_VERSION}
Error! echo
Your kernel headers for kernel 4.9.67-v7+ cannot be found at
/lib/modules/4.9.67-v7+/build or /lib/modules/4.9.67-v7+/source.
root@raspberrypi:/home/pi/rtl8814AU#