Get it on Steam... Should be pretty easy, I downloaded Steam (free) and stabbed at some random buttons that looked somewhat right.
When I added BeamNG to Steam, it was an instant add and as soon as I added it I could play... Weird?! Maybe I did something wrong?
I'm pretty sure you added a non-steam version of the game as a shortcut to your Steam. You should have recieved a Steam key for the game sometime in the past (through e-mail or here at the website). After you find it, you have to activate it through Steam.
Oh yes, now I remember, I did have a steam key. Thanks for reminding me, although you didn't plan too. I don't know if you have to buy Beam again though.