neither there is a direct way to do it in flutter nor any flutter package is available, but you can try direct android code.
try this :
String path2 = "/sys/block/mmcblk0/device"; BufferedReader serial; BufferedReader name ; BufferedReader manfid; BufferedReader oemid; BufferedReader mfDate; BufferedReader CID; serial = new BufferedReader(new FileReader(path2 + "/serial")); name = new BufferedReader(new FileReader(path2 + "/name")); manfid = new BufferedReader(new FileReader(path2 + "/manfid")); oemid = new BufferedReader(new FileReader(path2 + "/oemid")); mfDate = new BufferedReader(new FileReader(path2 + "/date")); CID = new BufferedReader(new FileReader(path2 + "/cid")); String sdSerial = serial.readLine(); String sdName = name.readLine(); String sdMfId = manfid.readLine(); String sdOemId = oemid.readLine(); String sdMfDate = mfDate.readLine(); String sdCid = CID.readLine();
or also use this
File input = new File("/sys/class/mmc_host/mmc1"); String cid_directory = null; int i = 0; File[] sid = input.listFiles(); for (i = 0; i < sid.length; i++) { Log.d(TAG,"sid info "+sid[i]); if (sid[i].toString().contains("mmc1:")) { cid_directory = sid[i].toString(); String SID = (String) sid[i].toString().subSequence(cid_directory.length() - 4, cid_directory.length()); Log.d(TAG, " SID of MMC = " + SID); break; } } BufferedReader serial = new BufferedReader(new FileReader(cid_directory + "/serial")); BufferedReader name = new BufferedReader(new FileReader(cid_directory + "/name")); BufferedReader manfid = new BufferedReader(new FileReader(cid_directory + "/manfid")); BufferedReader oemid = new BufferedReader(new FileReader(cid_directory + "/oemid")); BufferedReader mfDate = new BufferedReader(new FileReader(cid_directory + "/date")); BufferedReader CID = new BufferedReader(new FileReader(cid_directory + "/cid")); String sdSerial = serial.readLine(); String sdName = name.readLine(); String sdMfId = manfid.readLine(); String sdOemId = oemid.readLine(); String sdMfDate = mfDate.readLine(); String sdCid = CID.readLine();
//make sure before using this code check sdcard is present or not;
Above 7.0:Use StorageVolume.getUuid()on StorageVolume which you get from StorageManager.
The value is volume ID assigned during formatting of the card, and its length/format differs depending on file system type. For FAT32 it is XXXX-XXXX, for NTFS it's a longer hex string, for Internal mass storage, it returns null.

try this :
String path2 = "/sys/block/mmcblk0/device";
BufferedReader serial; BufferedReader name ; BufferedReader manfid; BufferedReader oemid; BufferedReader mfDate; BufferedReader CID;
serial = new BufferedReader(new FileReader(path2 + "/serial")); name = new BufferedReader(new FileReader(path2 + "/name")); manfid = new BufferedReader(new FileReader(path2 + "/manfid")); oemid = new BufferedReader(new FileReader(path2 + "/oemid")); mfDate = new BufferedReader(new FileReader(path2 + "/date")); CID = new BufferedReader(new FileReader(path2 + "/cid"));
String sdSerial = serial.readLine(); String sdName = name.readLine(); String sdMfId = manfid.readLine(); String sdOemId = oemid.readLine(); String sdMfDate = mfDate.readLine(); String sdCid = CID.readLine();
or also use this
File input = new File("/sys/class/mmc_host/mmc1"); String cid_directory = null; int i = 0; File[] sid = input.listFiles(); for (i = 0; i < sid.length; i++) { Log.d(TAG,"sid info "+sid[i]); if (sid[i].toString().contains("mmc1:")) { cid_directory = sid[i].toString(); String SID = (String) sid[i].toString().subSequence(cid_directory.length() - 4, cid_directory.length()); Log.d(TAG, " SID of MMC = " + SID); break; } } BufferedReader serial = new BufferedReader(new FileReader(cid_directory + "/serial")); BufferedReader name = new BufferedReader(new FileReader(cid_directory + "/name")); BufferedReader manfid = new BufferedReader(new FileReader(cid_directory + "/manfid")); BufferedReader oemid = new BufferedReader(new FileReader(cid_directory + "/oemid")); BufferedReader mfDate = new BufferedReader(new FileReader(cid_directory + "/date")); BufferedReader CID = new BufferedReader(new FileReader(cid_directory + "/cid"));

String sdSerial = serial.readLine(); String sdName = name.readLine(); String sdMfId = manfid.readLine(); String sdOemId = oemid.readLine(); String sdMfDate = mfDate.readLine(); String sdCid = CID.readLine();
//make sure before using this code check sdcard is present or not;
Above 7.0:Use [StorageVolume.getUuid()][1]on StorageVolume which you get from StorageManager.
The value is volume ID assigned during formatting of the card, and its length/format differs depending on file system type. For FAT32 it is XXXX-XXXX, for NTFS it's longer hex string, for Internal mass storage it returns null.

[1]: https://developer.android.com/reference/android/os/storage/StorageVolume.html#getUuid()