Motorola has offered Android 6 for the Moto e2 since February 2016. Why has Republic not offered an upgrade to that? Is it something specific with the Republic 2.0 plans? If I manually update it to 6.0 (by flashing the ROM) will it not work with Republic Services anymore?
The Moto E (2nd Gen) has a hybrid operating system unique to RW. There is no upgrade for this firmware. I believe it also has a locked bootloader and cannot be reflashed. If you were able to reflash it would no longer work with RW.
To the best of my knowledge, Motorola has not (and won’t be) releasing Android 6 (Marshmallow) for the Moto E2 in the U.S. It has done so outside the U.S. The relevant Motorola blog is here: https://blog.motorola.com/2015/10/02/marshmallow-and-smore-updated. Quoting with emphasis:
2015 Moto E with 4G LTE in Latin America, Canada, Europe and Asia_2 (2nd** Gen)
Unfortunately even if released in the US it would not be available for your RW E2 due to reasons billg mentioned as RW no longer supports Android system update for the legacy phones.