One thing to remember is that a number is divisible by 3 if the sum of all its digits is divisible by 3.Since 0 is not allowed and removing any one of the 4 digits results in a number divisible by 3,there're types of possibilities:
1.all 4 digits are the same,there're 9 possibilities:1111,2222,...,9999
2.all digits are 3,6,9:
(a) the 4 digits are 3,3,3,6:4 possibilities (take 333,then insert 6 to it)
(b) the 4 digits are 3,3,3,9:4 possibilities (take 333,then insert 9 to it)
(c) the 4 digits are 6,6,6,3:4 possibilities
(d) the 4 digits are 6,6,6,9:4 possibilities
(e) the 4 digits are 9,9,9,3:4 possibilities
(f) the 4 digits are 9,9,9,6:4 possibilities
(g) the 4 digits are 3,3,6,9:P₄/2 = 4/2 = 12 possibilities
(h) the 4 digits are 3,6,6,9:P₄/2 = 4/2 = 12 possibilities
(i) the 4 digits are 3,6,9,9:P₄/2 = 4/2 = 12 possibilities
(j) the 4 digits are 3,3,6,6:P₄/(2*2) = 6 possibilities (3366,3636,3663,6336,6363,6633)
(k) the 4 digits are 3,3,9,9:similar to (j),6 possibilities
(l) the 4 digits are 6,6,9,9:similar to (j),6 possibilities
Total:9 + 4*6 + 12*3 + 6*3 = 87