After some searching, it seems there are USB audio in ADC devices, none of which pass DC, voltage data loggers which do capture DC, but don't export to .wav files, and a few suggestions involving DIY platforms and ADC chips. Thank you all, by the way, for considering the question and the discussion related to sound cards. I'd rather not try to solve this by modifying sound cards though. At this time it seems the voltage data logger approach is the way to go. After requesting information, I was contacted by an engineer from one of the providers of voltage data loggers, and some of the devices have sampling rates up to 50 kHz, which should be adequate for a 44.1kHz wave file. The data is exported as a .csv file, which will require some code to assemble this into a .wav file, but I already have written code that does something similar.
James, you mentioned that some studio equipment is able to digitize DC. Any idea what that type of gear would be called?
Regarding the signals produced by the cycloid generators, since the images are not necessarily symmetric, recording these through capacitively coupled inputs results in the images' center of gravity, as it were, constantly drifting toward the origin, which does not look good. Also, these devices can function both as image generators, and offset generators. For example, the Digital Offset GeNerator (DOGN) board include effects such as ping-pong, random positions, and swooping motion paths which do not in any way perform as intended without the DC being preserved.
For those who haven't seen this yet, the following link is to a short video documenting the output from the DOGN code, in which it can be seen that there is DC which needs to be captured properly.
https://www.youtube.com/watch?v=fJ7kAR7D8yU