You can't use adb commands from inside the device, even if you somehow have it in the device you would need root permissions. The adb is the bridge between the PC and the device. Take a look here: https://developer.android.com/studio/command-line/adb
Although you probably can just remove the adb and use logcat directly, like:
Process p = Runtime.getRuntime().exec("logcat -d -v time -f /mnt/sdcard/logcat.txt");
Here you can see some more options about using the logcat command: Read logcat programmatically within application, including Reetpreet Brar's answer, that I think will be better for you:
Process pq=Runtime.getRuntime().exec("logcat v main"); //adapt the command for yourself
BufferedReader brq = new BufferedReader(new InputStreamReader(pq.getInputStream()));
String sq="";
while ((sq = brq.readLine()) != null)
{
//here you can do what you want with the log, like writing in a file to
//send to the server later.
}
Here you can choose methods to write your file: How to Read/Write String from a File in Android
Then, just send the file to the server.