I finally figured this out!
Add a Text Field to your form, Field Key is file. Set it to disabled (optional).
Add a button, Action is event, Field Key is selectFile, label is Select File.
Now mouse over the button and click the Create Trigger icon:

Name the form trigger and click Create:

In the Form trigger activity, choose “Select File clicked” for Event:

Add a Browse For File activity, output the path to SelectedFile:

Add a Set Form Values activity.

In the Dictionary, assign the SelectedFile value to the file object:

NOTE: If the Arguments field is locked/disabled, add a value to (or remove the value from) the Instance Name field. The change in this field seems to trigger unlocking the Arguments.
Now your Select File trigger looks like this:
Make sure you have a Run Local Triggers after you display the form:

Run your automation, click the Select File button, select a file (sometimes the window appears behind others and you have to look for it), and then click Open, and VOILA now you have a text box on the form that has the file path in it!

