Is there a way to have a field created on a form based on completion of another field or use of a button

It does not make sense in Quick Base to create new fields on the fly so I hesitate to explain how it could be done. 

It is just not good design to be generating new fields willy nilly.  It means that you actually need to have a child table to hold the data, so it's a design issue of the tables and relationships that we should discuss.

I prefer to discuss the use case and we will come up with a proper design for the tables and their relationships.