Cards that work
The Realtek RTL818x Linux drivers don't require distributions to ship any nonfree firmware to work.
It is unknown if they have or don't have any nonfree firmware inside.
Work to do to add support for more recent cards
More recent Realtek WiFi chip require nonfree firwmare. While the manufacturer didn't publish the firmware source code, it did publish the binaries under the GPLv2, and the GPLv2 allows reverse engineering, decompilation, etc, so it might be possible to reconstruct the source code somehow to get a free firmware.
Here's some pages with information about that:
- Group:Hardware/research/e-readers/Kobo/Aura_H2O_Edition_2 The Kobo Aura H2O2 WiFi driver has a GPL firmware
- In Parabola some people also found another driver with GPL firmwares that seems quite similar: https://labs.parabola.nu/issues/2680