Five Nights at Freddy's can exist within the Toy Story universe.
Throughout the Toy Story series, we see that the toys can come to life, from Buzz in the first movie and Buzz form the second, to Forky in the fourth movie. Any toy can come to life, regardless of how the toy came to be, and their minds when they first wake up consists of who they are and their basic instincts. However, just because any toy can come to life doesn't mean that the FNaF animatronics could as well, just that the little toys based on them could.
There are two main arguments against FNaF working in Toy Story:
1: The animatronics are never played with and therefore cannot come to life
2: The animatronics are large machines, not toys, therefore they would not be viable to come to life
For the argument that the animatronics haven't been played with, we've seen toys come to life that we can be sure haven't been played with. Stinky Pete from the second movie was mint in box and yet still came to life. Sure, he was capable of opening the box from the inside, but he says that he watched kids while he was still on the store shelf. Similarly, the aliens from the claw machine in the first movie were also alive despite being in a massive pile within the machine. Now, perhaps they were all played with at some point by the employees, but what about Zurg? His box was in a pile at the entrance to the store and he was functionally inert until his box got stuck in the doorway and he broke out. I am quite confident that Zurg here never got played with or activated until he broke out as he has the same fresh out of the box mentality and instincts that Buzz had in the first movie. Effectively, Zurg proves that a toy does not need to be played with to come to life.
As for the animatronics being large machines instead of toys? Well, if we only consider the movies, there likely isn't an example of what might be considered a large non-toy that comes to life. However, there is a perfect example outside the movies that would prove the FNaF animatronics could come to life: The Slimon minigame found within the Pizza Planet section of Disney's Activity Center, Toy Story CD-ROM. In this minigame, you are tasked with playing Simon Says against what the program handbook refers to as the Alien Slime Machine, which is 1) specifically a machine that dispenses slime (assumably a drink dispenser) and is not a toy, and 2) far larger than anything else that we've seen come to life, potentially being larger than an average adult if all four are considered one whole being. If these mechanical aliens can come to life, then the FNaF animatronics most certainly could as well.
Ultimately, there should be nothing that would prevent the FNaF animatronics from coming to life within the Toy Story universe. There's just enough evidence to support the idea without going off the rails and directly into severe head canon territory.
