How do I record a game clip without using a voice command?

As McFly said above, barring Smartglass, the voice option is all there is until October. When the October update launches to the public, by double-tapping the Xbox Home button on your controller, it brings up the "Snap Center" in which you will be able to
choose "Game DVR" to run snapped next to your active game.

Yes it is possible you can use the xbox smartglass app.

At the moment there is no possibilty to record a clip without the voice command, but in the october update for the dashboard a button command for the controller will be included.

For those of you who are having issues with recording clips and not having them actually save or for those of you who can't use Kinect voice commands, you can still record clips successfully using the Game DVR app. Unlike the "record that" voice command
feature that records up to the last 30 seconds only, the Game DVR app doesn't require voice commands and will record up to the last 5 minutes if you end a clip or up to 5 minutes if you start a clip. I've never had it fail to save a recording this way and
you're more likely to capture what you intended even if you wait for a significant amount of time (within 5 minutes). Try it!

You can record clips without using the Kinect - simply snap the game dvr app, choose to start or end a clip, the amount of time to record (up to 5 minutes), then it records. It's easy! You can't record a voice-over or show a clip of your face, but the ability
to record without the Kinect IS there. It actually surprises me how many people don't realize it.