How would I use a while loop to keep requesting user input

Use a while loop above input line as:

 while(true)

And, use if condition to break.

if(year == 0)
    break;

Also, condition for leap year is wrong in your code. It should be:

if((year % 100 == 0 && year % 400 == 0) || (year % 4 == 0 && year % 100 != 0))
    //its a leap year
else
    //its not

PS: As in comments, I’ll give a complete code:

import java.util.*;

public class Task10 {

public static void main(String[] args) {
    System.out.println("Enter a year to check if it is a leap year");
    while(true){
    Scanner input = new Scanner(System.in);
        int year = input.nextInt();
        if(year == 0)
            break;
        if((year % 100 == 0 && year % 400 == 0) || (year % 4 == 0 && year % 100 != 0))
            System.out.println(year + " is a leap year");
        else
            System.out.println(year + " is not a leap year");
    }
}

}