However, I am completely lost on what exactly the boards they are mounted on do. I have searched the internet but have come to dead end after dead end. Can anyone recognize these? (apparently I can only have one embedded image, but SIM900's board is similar):
SIM7600E-H:
Neither of these devices are in the table of hardware, why do you think openwrt supports them?
I suppose you can try soldering USB and power wires to the board and use it as an external modem with your OpenWrt router.