It is possible. But it may be difficult to get the alignment right. Also, when you create the field for the name, the width won't adjust automatically, so depending on the length of the name, there may be a big gap, or else it will be overset.
So, with those caveats, what you need to do is set up text fields in InDesign, and insert them in your text as inline objects. Now, make sure that all the text fields for a certain character have the same name in the Buttons and Forms panel. If multiple fields have the same name, when you export to Acrobat, you will find that if the name is changed in one field, all other fields with that same name will also be modified.
If you want to do it in InDesign and export the document after (without the possibility to change it later) you could use text variables in InDesign.