Why not "v létě praha je plná turistů"
Je would like to be the second word.
V létě je Praha plná turistů. or Praha je v létě plná turistů.
Does that mean "je" is considered a clitic?
Not in this case, where it is a form of the verb "být" ('to be'). Keep in mind that Czech is, despite its many quirks, mostly an S-V-O language, so let's not move the verb around too much.
I interpret the subject in this sentence to be "Praha v létě." We don't know if Prague in the winter is full of tourists.
I wrote "Praha v létě je plná turistů." Can you help me understand why this is not an acceptable form?
That is a possible sentence, but differs from the English original.
I believe it has the same meaning however. All we are saying is that the version of Prague that exists in the summer is full of tourists. The "in the summer" prepositional phrase doesn't need to be at the end in the english sentence.