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#Java 10, 140 139 bytes

v->{var r="  1\n x x\nx 9 x\n x x\n  x";for(int d=9,m=3;m<511;r=r.replaceFirst("x",d+""))for(;m==(m|=1<<(d*=Math.random()));d=9);return r;}

Port of @Arnauld's JavaScript answer, so make sure to upvote him as well!

Try it online.

Explanation:

v->{                // Method with empty unused parameter and String return-type
  var r="  1\n x x\nx 9 x\n x x\n  x";
                    //  Result-String, starting at:
                    //  "  1
                    //    x x
                    //   x 9 x
                    //    x x
                    //     x"
  for(int d=9,      //  Temp integer for the random digit, starting at 9
          m=3;      //  Bit-mask `m`, starting at 3 (binary 000000011)
          m<511     //  Loop as long as `m` is not 511 yet (binary 111111111):
          ;         //    After every iteration:
           r=r.replaceFirst("x",d+""))
                    //     Replace the first "x" with digit `d`
    for(;m==        //   Inner loop as long as `m` is unchanged after
            (m|=    //   `m` is bitwise-ORed with:
                1<< //    2 to the power
                   (d*=Math.random()));
                    //    a random digit in the range [0,9)
      d=9);         //     After every iteration: reset `d` to 9 again
  return r;}        //  And finally return the result-String