One way to see that it’s almost certainly wrong is to notice that it doesn’t in any way take into account the fact that there are $5$ vowels. On the other hand, the idea of starting with all $26^9$ strings and throwing out the ones with fewer than $3$ vowels is good. Specifically, you want to throw out the ones that have $0,1$, or $2$ vowels. (You went a step too far.)
There are $5$ vowels and $21$ consonants, so there are $21^9$ strings composed entirely of consonents $-$ i.e., with $0$ vowels.
To make a string with exactly $2$ vowels, you must choose which $2$ of the $9$ positions are to be filled with vowels, then choose vowels for those $2$ positions, and finally choose consonants for the other $7$ positions. You can do that in $\binom92\cdot5^2\cdot21^7$ ways.
How many ways are there to make a string with exactly one vowel?