No, There are no legal way for a 16-year-old to drive on public roads.
As Memor-X pointed out Wikipedia has a sentence about it.
Haruka is a racecar driver, even though she is barely sixteen years
old when she appears. However, the timing of her birthday to the
Japanese school year means she is one grade ahead of the Sailor
Soldiers.
And the Japanese version of Wikipedia has a annotation.
高校一年なのだが、制服姿でスポーツカーを愛車として公道で走らせている[4]。
She is first grade of high school, but she ride a sports car as beloved car.
4^ 国際運転免許証をもっていると考えれば合法であるが日本国内においては18歳までは公道では運転できない。北米版では留年しているという設定。
Legally she can drive a car with an international driver's licence, but in Japan, she can't drive on a public road until the age of 18. In the North American version, she has a scene where it's stated that she repeated a grade of school.
There is the Vienna Convention on Road Traffic.
Japan and US both signed this convention.
For example, I can drive a car in US with a Japanese driver's licence but without US driver's licence.
But the converse is little bit more difficult.
In Japan, the registration is required before driving a car.
(The convention says that the government can't have a test, but can have a registration.)
And Japanese government can't accept registration from people younger than 18 years old.
Thus a 16-year-old girl who has a US driver's licence can't drive a car in Japan.
Added: more research.
The author Naoko Takeuchi misunderstand the law; she thought a 16-year-old international driver licence holder can drive on public roads in Japan, and created the character and scene.
But the law doesn't allow it.
To fix the problem, in the North American version, Uranus was stated to be 18 years old.
For racer licence.
In Japanese law, there are no restrictions for driving inside private space.
For example, most Japanese get driver's licences in driving schools that have large space with fake road. You can get training to drive inside the school before getting your licence.
If the racing course allows it, children of any age can drive.
But (according to the Wikipedia article) most racing courses require racer licence.
Racer licence is certified by a company or agency (not the government).
The requirement is different between each company.
From Wikipedia's example, JAF (Japanese Automobile Federation) requires a normal driving licence to get a racer's licence, but MFJ (Motorcycle Federation of Japan) doesn't require a normal driving licence.
And racing course will accept international racer licences.
