In ArcGis desktop what I am trying to do is insert a coordinate (lat/long) on every township corner, I was wondering if this could be done by going to each corner and clicking to insert that coordinate.
If you have a polygon that represents the townships then you can convert those to points using "Feature Vertices to Point" in the Data Management Tools > Features toolbox and then use "Add XY Coordinates" tool to the points and then label the points using the XY fields.
You can also go the route of creating a Phython tool AddIn to give you the ability to click on the map and have it add a text graphic with the XY for the mouse click location (much more involved and requires programming skills).
There are a few options that I can think of.
If you just need to grab a few coordinates, you can add a graphical point, double-click properties, and click the location tab. (see picture below).
If you have a lot of corners, create a point layer, and you can add the XY to all the points automatically. Then you can use this to create labels or callouts.
If you already had the coordinated of the corners in the GIS layer you could also use the Add Label tool in the Drawing toolbar.