The TI-99/4A makes use of different kinds of storage media, which are also represented in MAME.
System ROMs
The read-only memory used in the TI console or in other members of the TI family are mask ROMs / EPROMs, or GROMs, a special kind of 8-bit streaming ROM with 6 KiB capacity. To be used in MAME, the contents of the ROM chips must be put into a file ("ROM dump"), and this file must then be stored in a ZIP container and put into a directory that is located in the "ROM path".
Peripheral device ROMs
Software modules
Also called cartridges, with ROM, RAM, or NVRAM
Cassettes
Floppy disks
Hard disks
RAM disks
External devices
Single files
This is probably a bit misleading, because MAME does, in fact, not support accessing single files. I thought it would be better to mention it so that you don't have to guess whether it works or not.