I am currently getting a house built and someone stated that you can call and have the county out to inspect the home as it is being built.

Is this typical/possible?

If so, who do I call to have the inspector out and check a few things out?

 

In most locations, Bldg Dept Inspections are mandatory at certain points in the construction. Check with your contractor and see when he gets inspections - IOW, when he calls for an inspection then you might want to be there when the inspector arrives

 

besides the official inspections that are required, you might want to think about additional "inspections" for your piece of mind- especially if you don't have much experience with building. It sounds like this is the case and that you might even have questions about your builder.
It could be an additional cost, but you might hire a qualified home inspector (usually used in purchase of a new home) or even another trusted builder/remodeler to do a couple of walk thru's with you. Even a friend in the construction trade. This would give you a second set of eyes- both to check up on things or to make suggestions. A friend of mine was having a new house built and had her trusted handyman do a couple of walk thru's with her. He found the builder to be up to code, but was able to make a couple of suggestions that she found helpful (getting computer network installed before sheetrocking for example.)