Jr. is a suffix. On a form it would go in a box called "suffix" and without that box it would be omitted or you risk creating a database error when entered.
Think of it not unlike the prefix Mr. or Dr., you wouldn't put "Mr. William" in the first name box - you'd likewise not use "Vandervilt, Jr." in the last name box.
A good form creator or database administrator would consider the importance of the suffix and include a box for it. Without it, you have to consider how the name would print in regular mail merge or form merge.
Since most of the time we use "Firstname, middlename, lastname" as fields to print in that order - if you wanted it to print correctly using just those three fields then the last name would have to be "Vandervilt, Jr."
This will create a sort order issue though. "Vandervilt, Jr." will come after all of the other "Vandervilt" records and you would have to remember that your name in this file is not "Vandervilt" but instead "Vandervilt, Jr." any time you needed to talk to someone about the database.