As far as I know, InDesign CC files will not open in CS6 due to new features
I don't use InDesign, so you will need to ask your client if InDesign CC has a "save as" option to save the file into the CS6 format
Or, you will need to subscribe to the Cloud version of InDesign Creative Cloud pricing and membership plans | Adobe Creative Cloud
If this is only short term need, be sure to sign up for a monthly subscription, not an annual subscription
InDesign does not save directly back to older versions, but can be saved as IDML, which can then be opened by older versions, The conversion is not perfect, though.
I suggest having the client work save as IDML, then open in CS6 (which they will have access to as part of the cloud subscription).
They need to check the file for accuracy, then save it from CS6. Going forward, they will need to work in CS6 for files you will be dealing with, or you will need to upgrade to CC
While you can do that, I have no issue telling that if I were the client I’m not so sure I’d be willing to accommodate you.
It’s on you to support the client, not the other way around.