The artist can send you a license, but an email will also work. Just make sure you print it as a pdf and save it for future reference.
The email must clarify what is included. For example:
Who owns the copyright? (Typically, the artist retains the copyright in the work unless you specify that the artist relinquishes their copyright. For example, if you don't get the copyright and full entitlements you can only use the model for the purpose you acquired it for, not further distribute it to make toys etc.)
Does your commercial license expire and is it unlimited? (For example, you are licensed to use this airplane model in as many projects as you wish, and to distribute as you wish for personal or commercial use.)
No further payments: The artist agrees that once the contract amount is paid they will have no further claim for monetary compensation or a share of the game.
And to protect the artist:
The artist doesn't warrant that the model is fit for use. (In other words, if you find the model isn't what you wanted, but is what you asked the artist to make, you can't claim your money back.)