I managed to get the SD card module working on ESP32! I created (based on another library peterfun/sd-card@1.0.6) a library that contains nodes for working with SD card on ESP32, and for example added nodes for working with string.
https://xod.io/libs/nazarijtipusak080/esp32-sd-card-read-write-append/
In order for SD card to be initialized and opened, it must be formatted as FAT16 or FAT32 (if it works with arduino, then everything is fine). IT IS IMPORTANT THAT THE FILE NAME MUST BEGIN WITH “/”. OTHERWISE OPENING OR CREATING THE FILE WILL BE A NODE ERROR. The example is tested on a regular ESP32. SPI PINS (MOSI -GPIO23, MISO - GPIO19, SCK - GPIO18) ARE INDICATED IN THE CODE. ESP32 MODELS OF DIFFERENT TYPES NEED TO BE CHECKED WITH THE DOCUMENTATION!!!