Can you install a sway bar on a car that doesn’t come with one?
Yes, kits are available.
It came with a front sway bar so are you wanting to put one in the rear?
Yes, you probably can. Whether you should or not is another kettle of fish. Be very careful when modifying your car’s suspension system, as there could be unintended consequences which could make the vehicle unstable in some situations. One of the posters here has a good deal of expertise w/suspension system design, maybe they’ll chime in on what to watch out for. If a reputable company is offering a sway bar they claim is applicable to a specific vehicle, that would greatly reduce the risk.
There won’t be any adverse haandling issues putting a 3/4 inch rear bar on an 86 Camry. The car can handle that, no problem.
A buddy and I prepped the suspension on a 92 4 cyl automatic Camry with 226K miles to see if it would work for a 24 Hr of LeMons $500 race car. Junkyard Camry 6 cylinder parts and some other tricks had the car cornering with the Corvettes at a track day test. Got smoked on the straights, but it only had 130 hp. Great fun. Never raced it tho…
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It might have cost more in fuel to trailer it to California for the race.
They hold them all across the country now, not just California. There are crazy racers EVERYwhere! 
So many crazies that there is ANOTHER crapcan race organization that used to be called ChumpCar now called ChampCar. A notch up from LeMons. They use a penalty lap system rather than a $500 cap.
The Camry owner and I whipped that 226K mile car all weekend for 12 - 25 minute track sessions and the old goat never missed a beat. Never overheated ANYthing. We adjusted tire pressure once by 1 psi. That was the only change. 85 was top speed with the straightaway we had but it cornered on rails. We added gas and a little oil.
The person did not say why they want a sway bar . I wonder if that is to solve worn struts on this vehicle .