Currently the only tag we have related to cartridges containing software in ROM is game-cartridge, with the following description:
For questions related to cartridge-based games and applications
and the following tag wiki text:
Use this tag for questions related to programs loaded from read-only memory cartridges.
Of the 31 posts with this tag, 7 or so are clearly not game-related at all in that they are not about cartridges for games nor about cartridges containing games. (These still fit under the description above, of course, because the description explicitly does not restrict the software to games.)
Some better options for the tag name (there may be others) are:
rom-cartridge: "ROM" is in this century frequently used with the meaning "software that was normally distributed in persistent memory not expected to be rewritten by the user," i.e., firmware. (MAME and other emulator users others often speak of "downloading ROMs," though of course that can't be done in the literal sense. And not infrequently, the images they download were originally on EEPROM anyway.) Nor, originally, was "ROM cartridge" ever thought to be restricted to cartridges containing no other components; most people would disagree that "there are few ROM cartridges for the Game Boy" because vast majority of them contain memory management and other logic, and even RAM.
software-cartridge: This makes it clear that the primary purpose of the cartridge is for delivering software. It also clearly does not restrict the software to being on ROM instead of, e.g., EEPROM or flash memory. Nor does it imply that the cartridge may not contain other logic, such as bank switching support. It might, however, be considered to apply to RAM-only cartridges, since those of course can also contain software if you put some on it.
firmware-cartridge: Perhaps makes it more clear that it contains software that's expected to be in memory that's persistent and not user-updated.
We may also want to consider how the naming will affect searches. If, e.g., someone types "ROM" into the tag field when they're assigning tags, what should and will come up? (All of the above will appear if someone starts typing "cartridge.")