Is the sentence, "I'm having my son wash my car", well said? The context is that the subject at that time has his son washing the car
As fixer1234 remarks in the comment, having someone do something is not perfectly clear with respect to the time element.
It could be an ongoing situation:
I'm having my son wash the car now that he's driving it too. He washes it every Saturday afternoon.
or a specific occasion:
I'm having my son wash the car before we drive to the airport to fetch his grandparents.
The tense alone, in and of itself, does not make a distinction between the two possibilities, and it can be used in either situation.
In both cases, the -ing form goes along with the idea of the ongoing. It is the son's ongoing responsibility to keep the car clean. And in the second situation, the son is either washing the car at this very moment, or it is the father's ongoing intention to have his son wash the car before they leave for the airport. We don't know which idea the father has in mind if we're not there to see whether the son is washing the car at that very moment, as might be the case if we're speaking with him on the phone. That kind of ambiguity is behind statements like the following:
I'm having him wash the car even as we speak.
He's washing the car at this very moment.
He's washing the car right now.
Your question is whether the statement is well said.
No problem in what you said, but common structure that I know is - you get done something by someone. The order changes there.
I'm getting my car washed by my son.
Or to remain polite, I may opt for:
My son is washing our (I won't be bossy by calling it 'my') car as I asked (again, I'd be polite 'asking' him rather than ordering him by 'telling') him to do so.