To edit vertices of a polyline in AutoCAD, you can use the PEDIT command or grip editing.
Easier, interactive adding (inserting) of new vertices, or deleting (removing) existing vertices can be done with the help of the free LISP utility CADstudio AddV. With its commands AddV and DelV you can add or delete a polyline vertex just be picking a point with your mouse cursor.
There are two versions of the "add" and "del" commands - with a single click for select and add/delete (AddV1, DelV1) and with a separate select and click to add/delete (AddV, DelV).
You can download the AddV utility from www.cadstudio.cz/download, load it with APPLOAD.