Just took a look on my phone, so not the greatest view.
1) You have a ton of comments. It almost makes the code unreadable.
#User input variable
self.user_text = ""
#import morse dictionary
self.morse = morse_code
this is an example of what I'm talking about. You have already imported the morse_code library, here you are creating an object for it, so the comments actually add confusion.
Python prides readability over anything.
2) I know Pygame is a popular library, but I'm not as familiar with it, looks like a lot of the stuff you are doing should be in a UI class.
Personally, I would use pyqt, it is my go to for python UIs. Using QT Designer solves a lot of the headache you are doing there.
I am happy to pull your branch and give you some help or show you an alternative way. What you have so far is pretty decent.
Let me get this straight tho, you want an input box where the user types a sentence, words, whatever. Then you take those letters and retrieve the corresponding Morse value, then display that back to the user. Correct?