This depends greatly upon which "government organization" you work for. Most of the answers here are assuming a commercial SSL. Some government organizations allow that, but most require a .gov or .mil domain. So those will not work. Each government organization has their own regulations. Typically, your parent organization will need to authorize you to have an Intermediate CA and may be the one to issue it to you. Perhaps ask the people who issue your normal SSL certificates. They probably don't know nor have any useful answers, but they may be able to direct you to who does. This may involve several people and getting approval from relatively high ranking people (who may or may not actually understand what you are asking for).
You'll need to have a VERY good explanation of your intended use case and what you will use it for. You may need to produce official policies for your organization to limit what can be done with your Intermediate CA to make them feel more at ease. If they are bound by certain regulations, you may need to find those regulations and determine the exceptions or waivers that are available. Once you determine which one may apply to you, you may have to explain it to them. If they do not understand, be patient and polite, and keep asking. You may be able to speed things along by getting someone in authority over them to authorize it (this takes the responsibility off their shoulders, and may allow them to not care as much). You take the risk of pissing them off if you "go over their head" though. So, ask them about it first (some thing like, "If I asked your 'supervisory organization here' to authorize it, would you be more comfortable?").
Often times this is a very long battle and an improbable task (not impossible, but may be highly improbable depending on your organization). This is especially true in Department of Defense (or military) organizational structures.
There may be better routes to achieve the same thing. You could request a wild card certificate and then make the branches sub-domains. However, sometimes getting a wild card certificate is just as improbable. I would start with whomever you get your normal SSL certificates from and work your way up.