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.

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

In the National Guard the answer is yes if there is a deploying Guard unit that has a shortage in your MOS.

You can volunteer for anything but yes, if there is an available slot with you are the correct pay grade you can volunteer.

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.

Wait until you get your first one down, then think about it. Depending on your branch and MOS, some deployments can be rather rough. A much needed break in between is a blessing before you step off for the next one.

Always, the level of difficulty in execution commiserates with the correlating component as well as "outside your realm of control" issues that may arise.