From the ATtiny85 datasheet:
7.1 Idle mode
When the SM[1:0] bits are written to 00, the SLEEP instruction makes the MCU enter Idle mode, stopping the CPU but allowing Analog Comparator, ADC, USI, Timer/Counter, Watchdog, and the interrupt system to continue operating. This sleep mode basically halts clkCPU and clkFLASH, while allowing the other clocks to run.
Essentially, the CPU stops executing instructions in idle mode. So writing to EEPROM (which requires instructions to be executed) is not possible.
If the CPU gets woken up by one of the stated sources, then you can read/write to EEPROM (or do anything else) and go back to sleep.
I'd recommend reading through all of chapter 7 in the datasheet if you want to minimize power usage. There are quite a few options.