One of my friends has started using the word 'vying' more and more these days. He did not know the word before a certain date and there was a clear event which caused him to "acquire" it into his lexicon. Is there a formal term for this phenomenon? What is it called when someone suddenly starts using a word? English is his first language.
It’s called ‘reconstructability’ of language (Russian достраиваемость языка, the term of S. Burlak). In a nutshell, nobody remembers every single word of a language. For example, one may be acquainted with the verb ‘vie’ but he or she has never heard any forms of it. If one is a fluent English speaker, then one is pretty well aware of the model of constructing -ing forms, e.g. pick, bite, invent get -ing in contexts like “He is ___ my door.” It doesn’t matter whether one has ever pronounced ‘vying’ or not—all the mentioned factors make one capable of forming ‘vying’: “They are vying in swearing at me.”
This works with almost every other word: if there is an information about how to build -ing (plural, -ed, etc.) forms in one’s mind, one will be able to do it at any moment.
P.S. The previous version of this answer contained the description of nonce words and neologisms as they are really close to the topic. The main difference between the phenomenon in question and neologisms and nonce words is that the latter are used to describe brand new words.