Is it possible to find any information about a proprietary IC with no manufacturer name on it?
No. Applications support is costly, especially for microcontrollers. With a house numbered device, most likely this support is provided directly to the OEM (Original Equipment Manufacturer, Kenmore) by a specific account manager contact at the chip vendor (let's say Hitachi for example). Likely it's based on some publicly visible microcontroller like maybe Renesas H8 (wild guess, I could be way off), possibly customized for that large OEM customer. Since the NRE costs (Non-Recurring Engineering) are paid by the OEM, that design belongs to the OEM. Bear in mind that Kenmore has competitors. If OEM 1 specs a custom chip that gives them an advantage like lower cost or more features, they won't make it easy for competing OEM 2 to benefit from their design.

Legions of corporate lawyers protect this kind of proprietary intellectual property information, which is worth millions of dollars of NRE and untold opportunity costs.

Your best bet is to search on "vacuum fluorescent display controller" and select something publicly supported. (Disclaimer: I'm an applications engineer at Maxim Integrated, and apparently several of our display driver chips are near or at the top of this google search.) You may find either display driver chips that require a user-provided microcontroller, or possibly a microcontroller with integrated VFD drivers.