123, 132, 321, 312, 213, 231. There are no 2 digit numbers by SFFT. There are no 4 digit numbers by intense casework. There are no 5+ digit numbers because the product must be at least n! and the sum must be under 9n, and we have $n!>9n$ for all integers greater than 5. QED.

no need for intense casework, if a + b + c + d = abcd are all distinct then a+b+c+d = abcd >= 24 implying say a >= 6, but then a+b+c+d = abcd >= 36, absurd since a+b+c+d <= 9+8+7+6