Yes, especially in the peak travel months, June through August, tickets do sell out (sometimes months in advance), and you should book earlier rather than later.
If you’re interested in experiencing the Vatican while in Rome, we recommend touring with Get Your Guide because they have a multitude of ways you can experience this attraction (self-guided, audio-guided, historian/expert-guided, etc.) Their tickets are also often cheaper than those of the main museum’s website, which you can check out here.