Is there an ADC chip that reads only when triggered that can be synchronized with PWM?
You should change your architecture some. There are plenty of MCUs that have A/D converters right on board. You can simply feed the triggering signal into the MCU to wake it up. When the MCU wakes up it enables the A/D to take a reading. When the reading has been processed the MCU disables the A/D converter again and puts itself back to sleep to await the next trigger.

This should eliminate the need for two separate chips with some slow serial data interface between them. 

Unless there is some aspect that you have not told us about this would be a better overall solution. 

There are some applications where a separate A/D may be needed such as very high resolution converter, or one that has to work at very very low noise levels where you isolate the converter from noisy digital circuits.