Like I said, nothing.
I believe the Tolino and similar lesser-known brands do not operate a store at all -- they require you to sideload EPUB ebooks purchased from e.g. Kobo, B&N, GooglePlay, and direct from publisher.
No sure if they allow you to sync at all.
As for the main brands -- the ones that
- Offer an E-Ink device
- Operate an integrated store
I am pretty sure that is just Amazon (Kindle), Kobo (Kobo devices), and B&N (Nook).
Sony is out of the business.
Your best bet was Amazon, and they don't sync with the Desktop.
You will have to look at third-party applications. Like KOReader, which if you add Ubuntu Touch into the mix might actually be a complete solution. (Not sure where Ubuntu are holding these days, but I'm sure they must have preview stuff at least which runs the phone bits on a desktop.)