All sales of handguns across state lines MUST go through an FFL. That means an Indiana FFL must ship to the buyers Illinois FFL. No exceptions.
It entirely depends on the FFL. While some won't accept transfers from private individuals, a great many FFLs will do so without any problem. It's perfectly legal and perfectly normal. I usually have used "home-based" FFLs who run it as a side business out of their homes and usually they have no problems with receiving a firearm from an individual, charge a very small amount, and are great to deal with.With the current administration working to shut down FFL dealers, most dealers will not accept a transfer of a handgun across state lines unless it comes from an FFL.