If one allows non-leading zeros then there will be infinitely many solutions. Even if one excludes those, the count will be huge. I restrict further by enforcing that digits be (non-strictly) increasing. One can get all possible values from these solutions simply by reordering and, if desired, inserting zeros.
The idea is to use Solve, giving restrictions on the variables as noted above, and using the Integers domain setting. Solve will then use integer linear programming under the hood and cough up a solution set in reasonable (I think) time.
n = 36;
vars = Array[x, n];
AbsoluteTiming[
soln = Solve[
Flatten[{Total[vars] == n, Map[0 <= # <= 9 &, vars],
Table[vars[[j]] >= vars[[j + 1]], {j, Length[vars] - 1}]}],
vars, Integers];]
Length[soln]
(* Out[71]= {32.0909, Null}
Out[72]= 7657 *)
Check the first and last ten.
10^Range[0, n - 1].vars /. soln[[1 ;; 10]]
10^Range[0, n - 1].vars /. soln[[-10 ;; -1]]
(* Out[73]= {111111111111111111111111111111111111, \
11111111111111111111111111111111112, \
1111111111111111111111111111111122, \
111111111111111111111111111111222, 11111111111111111111111111112222, \
1111111111111111111111111122222, 111111111111111111111111222222, \
11111111111111111111112222222, 1111111111111111111122222222, \
111111111111111111222222222}
Out[74]= {225999, 135999, 45999, 1116999, 126999, 36999, 117999, \
27999, 18999, 9999} *)
