I know for a fact that SDCard (SDIO) support is available on the PSoC5LP chips using the emFile middleware component.
SDCard support is available for some of the PSoC6 chips. There is a example project in Modustoolbox called "USB_Mass_Storage_FileSystem". This is available for the Cypress kit CYPROTO-062-4343W and other kits.
In both cases, the emFile middleware SW allocates SPI resources and provides API calls to handle file/directory creation and read/write access.
Can you please elaborate on the requirement? We have PSoC6 chips that come with SDHC controller through which you can implement the SDIO interface. But note that SDHC doesn't support SPI protocol mode inherently.
But here is a reference you could use where a user created a project that implements a SPI based SDHC driver: emFile component for PSoC 6