It would be easier to count the number of words that have $2$ or fewer vowels, and then subtract this number from the total number of words (which you have already computed).
I assume that by "$3$" or more vowels" you mean $3$ or more occurrences of vowels, so in particular a word with 2 e's, 2 i's, and the rest consonants qualifies.
How many words are there with no vowels? Clearly
$$21^7$$
if, as per usual convention, we agree that there are $5$ vowels.
How many words with $1$ vowel? Where the vowel occurs can be chosen in $\binom{7}{1}$ ways. For each of these ways, the vowel can be chosen in $5$ ways. And once you have done that, the consonants can be filled in in $21^6$ ways, for a total of
$$\binom{7}{1}(5)(21^6)$$
Finally, how many with $2$ vowels? The location of the vowels can be chosen in $\binom{7}{2}$ ways. Once this has been done, the actual vowels can be put into these places in $5^2$ ways. And then you can fill in the consonants in $21^5$ ways, for a total of
$$\binom{7}{2}(5^2)(21^5)$$
Add up the $3$ numbers we have obtained, subtract from $26^7$.
Our argument was a little indirect. We could instead find, using the same sort of reasoning, the number of words with $3$ vowels, with $4$ vowels, with $5$, with $6$, with $7$, and add up. This is only a little more work than the indirect approach. But any saving of work is helpful! Also, the indirect approach that was described lets us concentrate on pretty simple situations, the most complicated of which is the $2$ vowel case.
Remark: The calculation we have done is closely connected to the Binomial Distribution, and if you have already covered this, it may be the point of the exercise. So if you know about the Binomial Distribution, imagine the letters to be chosen at random. Then the number of patterns with $3$ or more vowels is the probability of $3$ or more vowels, multiplied by $26^7$.
Let's break it down.
The first letter obviously has 26 choices. The second letter can be any of the letters, minus the previous one, so 25 choices. The third letter can be anything except the second letter, so 25 choices, and likewise for every other letter.
Thus the number of n-length words of this type is $26\times25^{n-1}$, or in this case:
$26\times25^{4} = 10156250$.
Best Answer
If no A is used, there are $29 - 1 = 28$ ways to fill each position, so there are $28^5$ possible five-letter words that do not contain an A.
Your answer $$P(28, 5) = \frac{28!}{(28 - 5)!} = \frac{28!}{23!}$$ would be correct if letters could not be repeated.
The set of five-letter words in which the letter A is used at least once is the complement of the set of five-letter words in which the letter A is not used once. To find the answer, subtract the number of five-letter words in which the letter A is not used once from the number of five-letter words that can be formed from the alphabet.