How can I set an API key once and save it within a function?

You can save them as properties using $this

I'm not sure if you want to reuse the client or the app id/key, but it's pretty much the same idea either way.

// Class example
class Airport
{
    private $appId;
    private $appKey;
    private $client;

    public function setClient($appId, $appKey)
    {
        $this->appId = $appId;
        $this->appKey = $appKey;

        $this->client = new GuzzleHttp\Client([
            'headers' => array(
                'resourceversion' => 'v4',
                'accept' => 'application/json',
                'app_id' => $this->appId, // Set this
                'app_key' => $this->appKey // And this
            )
        ]);
    }

    // New function that uses the client
    public function someOtherMethod()
    {
        $x = $this->client->someMethod();
    }

    // new function that uses the app properties
    public function anotherMethod()
    {
        $x = new Something($this->appId, $this->appKey);
    }
}