ELM Petit FAT File System Module claims:
It can be incorporated into the tiny microcontrollers with limited memory even if the RAM size is less than sector size.
In fact:
- Very small RAM consumption (44 bytes work area + certain stack).
It is open source; so you might take a look at how it does that (or just use it as-is).
The write function does have some significant restrictions, such as while it can modify a file, it cannot create or change the size of one - which is probably the compromise necessary to avoid sector buffers.
There is a full featured version of ELM FatFS, but that requires significantly more code and RAM space.
The page also has links to the FAT32 specification, and some technical notes on how SD cards work which might be useful.
EDIT:
In fact this is not particularly helpful. Petit FAT's claim refers only to the filesystem layer itself, it does not include any hardware specific device driver, and for SD/MMC there is no getting away from the 512 byte read-modify-write cycle.
Although the AVR may not support memory-mapped external memory, one could use an external serial memory device such as a Cypress FRAM or nvSRAM to store the sector data; although being non-volatile, it may remove the need for SD altogether unless large removable media is specifically required.