Can we introduce pivotal characters part way through the book? Yes. Pivotal characters can be introduced almost anywhere in the book with the exception of the end. A character can be repeatedly reference but not make a physical appearance until the end but if a pivotal character turns up at the end it all seems a bit Dues Ex Machina, God in the Machine.
The concept of ‘God in the machine’ is particularly associated with the theatre of ancient Greece where The Gods often intervened at crucial points in the story, particularly towards the end. However, in modern fiction the idea of a character randomly appearing to alter the ending of a story is rarely accepted by a reader. This is perhaps due to the change in expectations over the centuries, a move away from the idea of Gods directly intervening in the lives of mortals but whatever the reason people prefer to have a chance to anticipate the outcome. By this I don’t mean they can predict how it will end but rather a sense at the end that the clues were all there even if they didn’t catch them all. Pivotal characters appearing mysteriously at the end can seem like the writer is cheating the reader.
Now, while they can’t suddenly appear at the end although the beginning of a story is generally associated with introducing the characters they can appear later on in a story. For example in The Fellowship of the Ring, it’s a fair way into the book that some of The Fellowship are introduced. In Jane Eyre we’re ten chapters into the story before we meet Mr Rochester, the enigmatic hero. In a murder mystery we usually meet the murderer in person but not always at the beginning.
When we introduce a pivotal character later in the story we don’t have to draw attention to the fact they will be important they can, if we wish, slip in and gradually come to the fore. An example might be The Hound in Game of Thrones/A Song of Fire and Ice where he begins as a minor reoccurring character and becomes an important one a few series/books in (carefully trying to avoid spoilers). By gradually building The Hound George RR Martin also has a chance to give the reader an initial impression of the character which proves to not be entirely accurate. This doesn’t have to be built up over several books and can be done in a single one an obvious example would be the romantic cliché of the heroine and hero meeting and hating each other but falling in love by the end. This can be done by playing with how much information we give the reader, an example being of the way The Hound becomes, to a degree, a more sympathetic character the more we learn about his backstory and through the development of his relationship with Sansa Stark which shows he isn’t the heartless character we first assumed.
So simply because the beginning is associated with introducing all the key players in the story they don’t all have to be revealed at the beginning. And, even if they appear later in the story, they don’t have to reveal their true colours quickly.