In the movie Iron Man 2, Tony Stark figured out how to synthesize Vibranium. Why didn't he use it to make a new Iron Man armor?
Because synthesizing Vibranium still required a source of the raw extraterrestrial ore. There was an implied limit to the availability of Vibranium in the Marvel Cinematic Universe.
Captain America's shield was supposed to be all the free Vibranium in the world (Howard Stark says this and we can assume he was an authority on the subject.)
- Until relatively recently, the Wakandans lived as their ancestors did, primarily a hunter-gatherer society with some cultivation of crops. Central to the Wakandan's myth and ritual was a small mountain of ore which according to legends fell from the sky. This meteoric ore was later found to be Vibranium, an extraterrestrial metal capable of absorbing all sonic vibrations. -- Wakanda, Marvel Wikia
In the canon comic universe of Marvel Earth-616, most of the world's Vibranium was found in the hidden kingdom of Wakanda, the home of the Black Panther. Since Wakandan Vibranium is theorized to be an extratrerrestrial material, this may explain why there is a limited amount of the metal on Earth.
The Antarctic Vibranium (also known as anti-metal) variant of Vibranium destroys metallic bonds and can only be found in the Savage Land.
Captain America's shield is comprised of an Adamantium/Vibranium/Uru alloy, making it one of the most indestructible objects on Marvel Earth.
It has not been determined that anyone has been able to create an artificial (i.e. synthesized) version of Vibranium in either the comic or the Marvel Cinematic Universe.
One attempt to create an artificial Vibranium created a new material called Reverbium. After attempts at weaponizing it failed, all of the material was presumed destroyed.
Reverbium is an artificial variant to Vibranium created by Sanjani Jaffrey, after most of the world's Vibranium was rendered inert. This variant is imperfect, so rather than absorbing vibrations, it does the opposite, and blows everything away, making it a powerful weapon even in small amounts.
The canon comics Iron Man uses Vibranium as a control substance replacing the Palladium core in his Arc Reactor.
The Vibranium Mound of Wakanda has been destroyed by the Black Panther to prevent Doctor Doom from gaining control over it.