OK, I have made some progress. I will leave this here in case someone is struggling with the ADC, chains and triggers.
This application note for the rt1020 is extremely informative, with diagrams and sample code: https://www.nxp.com/docs/en/nxp/application-notes/AN12200.pdf
The sample code for this application note is contained in the Windows executable package available here: https://www.nxp.com/docs/en/nxp/application-notes-software/AN12200SW.exe
Specifically this diagram helped a lot:

So to answer my own question.
The ADC_ETC sync mode should be set true on trigger4 (not trigger 0). When trigger 0 is executed, trigger 4 will also execute.
Trigger 0 is tied to ADC1, trigger 4 is tied to ADC2. So you would set up both triggers 0 and 4, then you would also set up chains for trigger 0 and trigger 4.
Since I only need a single conversion, my chain length is 1 for both triggers. Both triggers need a configured chain 0 element, with the conversion channel for ADC1 set up on trigger0, chain0 and ADC2 channel on trigger4, chain0.
I fire the Done0 interrupt on trigger0,chain0 and the Done1 interrupt on trigger4, chain0. Then I know both conversions are complete when both interrupts have fired.
The ADC peripherals need to also be configured. The one specific setting I found was that ADC_EnableHardwareTrigger should be true for both ADC, although I was expecting this to be false using a software trigger. The channel numbers don't seem to matter, because they are set explicitly when the chain element executes (with it's own channel config). The interrupt on conversion complete should be disabled, since we rely on the chain interrupt.
To read the conversion results, you need to read the appropriate trigger/chain register from ADC_ETC.
So in this case, trigger0/chain0 "ADC_ETC_GetADCConversionValue(ADC_ETC, 0U, 0U)" in the Done0 interrupt
and trigger4/chain0 "ADC_ETC_GetADCConversionValue(ADC_ETC, 4U, 0U)" in the Done1 interrupt.
Hope that will help someone in the future!