Is it vapor locking or not? All else shows okay.

doubtful that vapor lock is an issue on a fuel injected car. it was something that happened on old vehicles, and you just don't hear about it any more. It doesn't happen because of the high fuel pressure in the fuel lines of fuel injected vehicles. If the vehicle is cranking and won't start, it might be a fuel issue--getting no fuel because of a fuel pump issue. But not vapor lock.

Vapor lock can be caused by many different things, all coming down to the same root cause: too much heat in the fuel feed lines.

Low fuel pressure can cause the amount of heat needed to vapor lock to be lower than normal, so checking fuel pressure is important. Fuel pump going out feels a lot like vapor lock.

Engine overheat can also cause vapor lock. Clogged fuel filter can cause vapor lock. The fuel line should be routed away from the exhaust system.

Check cankshaft sensor.

If it is a gas engine, its almost impossible to get vapor lock.
Vapor lock is when liquid gasoline turns to a vapor which will not flow thru a carburetor. The usual cause was hot fuel lines going to the carb.
Since the early 1980s engines have used fuel injection which uses liquid fuel under pressure.

Vapor lock occurs when the gas in a fuel line turns from liquid to vapor, before it reaches the engine. The first signs of vapor lock are normally when your engine begins to run roughly. The engine may sputter and hesitate. It will then stall completely and refuse to restart. It may seem like the car is out of gas or the fuel pump has failed. While these are possibilities, vapor lock may be the problem.

Fuel injected engines cannot get vapor lock. First you need to explain how you came to the conclusion if vapor lock. But this is an old post and probably will stay dead.

No.
To begin with, there is no such thing as vapor lock in the cooling system. Vapor lock is when liquid gasoline vaporizes due to excessive heat. If the engine got hot enough to turn coolant into vapor the engine would seize up.
Do you have trouble code numbers ?

If you are serious about 'vapor lock', then this symptom will/would show up during a fuel pressure test/check. Did they check this? The fuel pressure on the 2003 Elantra is 350 kpa (3.5 kg/cm, 49.8 psi). At this pressure, there would be NO vapor lock. Now, if the fuel pump is 'INTERMITTENTLY' dropping in fuel pressure, it will 'fuel-starve' the engine & the engine management fuel ADAPTIVES will show a shift in the fuel trims (lean condition) by adjusting the fuel trim adaptives to + (more fuel) by keeping the injectors open longer. However, this may not be the problem. There are numerous systems which could be the cause, OR two or more systems contributing to the problem. The best advice is to have this condition properly diagnosed. Since you've taken it to 3 different dealers already, perhaps you should find a 4th.

Check the plugs.

when they went to fuel injection the fuel that was not used returned to the tank thus cooling the fuel and ending vapor lock, my question is did you do a fuel pressure test? and di you do a fuel leak down test? and the first thing I would do is change the fuel filter. with ethanol in the gas it kills the filter. I have to replace mine every three months they just plug up from that junk. Other than that the only other thing would be a coil breaking down when it gets hot the coil shorts out when cool it starts so check those things first. be sure fuel pressure is 50 to 60 PSI and does not drop below 30 PSI for at least a minute, if it does the fuel pump check valve is bad, also the fuel sock could be bad as well.