Can abbreviations become actual words?
Clearly, yes. The word "cab", for example, was originally short for cabriolet.
and is there a process for this?
Not a formal one. Something in English becomes a "word" when enough people use it as such.
There is no "board of officially approved English words" to which you have to submit a request.
Words in wide usage eventually find their way into dictionaries, but remember that modern English dictionaries are (primarily) descriptive, not prescriptive:
The truth of the matter is that today virtually all English language dictionaries are descriptive. The editors will usually say that they are simply recording the language and how its words are used and spelled. True, there may be some guidance. For example, most Merriam-Webster dictionaries will note if certain words are deemed nonstandard or offensive by most users; however, the words are still included. Of modern dictionaries, only the Funk and Wagnall's contains a certain amount of prescriptive advice. All the major dictionary publishers - Merriam-Webster, Times-Mirror, World Book, and Funk and Wagnall's - will tell you that they are primarily descriptive.
So if some dictionaries differ on whether to include certain words (mini, temp, stat), that is likely as much a reflection of the dictionary's editorial policy than it is a ruling on whether something is "really a word".
For example, here is Merriam-Webster's FAQ on how a word gets into their dictionary:
To decide which words to include in the dictionary and to determine what they mean, Merriam-Webster editors study the language as it's used. They carefully monitor which words people use most often and how they use them.
They describe a process by which editors scour print and electronic publications for new words or new word usages, and track them via computer. Eventually, these "citations" might add up:
Before a new word can be added to the dictionary, it must have enough citations to show that it is widely used. But having a lot of citations is not enough; in fact, a large number of citations might even make a word more difficult to define, because many citations show too little about the meaning of a word to be helpful. A word may be rejected for entry into a general dictionary if all of its citations come from a single source or if they are all from highly specialized publications that reflect the jargon of experts within a single field.
To be included in a Merriam-Webster dictionary, a word must be used in a substantial number of citations that come from a wide range of publications over a considerable period of time. Specifically, the word must have enough citations to allow accurate judgments about its establishment, currency, and meaning.