I have a party idea where a group of eager to learn young apprentices leave their masters tower after getting impatient with his method of teaching. But they only remember to take one spell book, and they just decide to go with it. Are there any rules that prevent this? Or any issues that could potentially arise from a situation like this?

They can share the book, but not the spells

If a wizard has another wizard's spellbook, they can't actually prepare the spell until they scribe it into their own spellbook. I don't see any real mechanical reason why you couldn't use a single spellbook for that purpose, but every apprentice would have to scribe their own copy of a spell in a way that they can understand and quickly memorize it.

Copying that spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.

So your spellbook would end up with one spell as scribed by Apprentice A, who can now prepare that spell. The other apprentices can't prepare their spell from that entry, but they can copy that spell into their own scribed version of that spell.

So your spellbook now has the same spell twice, and two different apprentices can prepare from it, but the other ones still can't do it. So it's not exactly the most effective way of storing your spells, because that spellbook is going to be a mess of different notations.

Imagine sharing one character sheet with 4 different characters on it with your party, and you'll see where this will become incredibly frustrating for your wizards.

As Gorp noted in the comments, this will result in your spellbook getting full pretty darn quickly, too.

Want to share spells?

Comments suggested that it wouldn't really be that big a deal if all wizards could use the spells from the book without scribing their own version into it, but I disagree, at the very least if this is a player party concept. A wizard is extremely flexible and learns a lot of spells already, if you allow a group of wizards to combine their level-up gained spells into a single book, and everybody benefits from it, then you've essentially given every wizard extra spells based on how many other wizards are in the party. This can easily lead to a party of wizards that has almost every spell of their level at their disposal. It's not the end of the world, but it's going to make them a lot more flexible, as now they can pick up all the utility spells and still take all the big nuke spells.