Raquel uses six different digits to fill in the blanks below, writing one digit in
each blank, so that the resulting addition statement is correct. What is the least
possible sum of the six digits?
Let's consider the possible ranges for each digit.
Since the maximum sum of two digits is $17$, the hundreds digit must be at most $1$ and the thousands digit must be at most $9$ (since there must be carrying).
Since the hundreds digit is at most $1$, it must be $0$.
Since the thousands digit is at most $9$, it must be at least $5$ (because there is carrying).
The remaining four digits are at most $4$, so the maximum sum of these four digits is $4+4+4+4=16$.
The smallest possible sum is $5+16=\boxed{21}$.