Can you volunteer/ request for more deployments?

Jordan,
You can always volunteer to deploy. But, I can tell you from experience that you're looking for career progression/promotion, don't get stuck doing deployments. Experience wise, deployments are good because they provide you with countless situations which will develop and become a good solid leader. But you have to balance everything out, meaning spend time in garrison so you can experience other things/situations, go to school, etc.
I always liked to deploy, but found out a little too late that serving and performing in garrison matters.

PV2 J M Be careful what you wish for. You can volunteer for anything, but there is a time and place to do that. Make it through training and your first deployment and you will have a much different outlook. Combat is not like a game or the movies. You will be forever changed, so your mission is to take it one step at a time. Carefully plan, set goals, implement.

Make it through Basic Training and AIT/OSUT then worry about deployments