Sevens inside sevens

The digit sum of a number below $1000$ is at most $27$, so only a few digit sums are possible.

1. A number from $1$ to $999$ has digit sum at most $27$, and $1000$ has digit sum $1$, so the digit sum must be $7$, $14$ or $21$.
2. Among multiples of $7$ with digit sum $7$ we find $7$, $70$, $133$, $322$, $511$ and $700$, which is $6$ numbers.
3. With digit sum $14$ we find $77$, $266$, $329$, $392$, $455$, $518$, $581$, $644$, $707$, $770$ and $833$, which is $11$ numbers.
4. With digit sum $21$ we find $399$, $588$, $777$ and $966$, which is $4$ numbers, so the total is $6+11+4=21$.

Answer: $21$