How do I create a button within a Quickbase form that allows users to add fields when needed?

There are multiple way to handle this depending on your goals, here are a few options:

A) You may manage the access of each Role to manage your table schema via the "Edit Field Properties," but, that's something typically managed by Admins.

B) Perhaps you are asking how may a User add "options" to an existing dropdown, which is handled via the field's Properties where you check the box to "Allow users to create new choices." However, that option is not available for all field Types, but is available for "Text - Multiple Choice" fields. Then, when a User is in edit mode and they click the dropdown, there is an option at the bottom to add additional choices.

C) Or you could create a one-to-many relationship to a table which houses options as records and allow User's to add records into this table.

Likely, you are hoping for "B" where you check the box to allow users to create new "choices" (not fields) on an existing dropdown. But, a word of caution is those options can get messy fast. See screenshot!