Jeremy Crawford unofficially addresses this specific issue regarding Lost Mine of Phandelver in a pair of tweets from January 2017:
in LMoP, if you kill a wizard (i.e. Black Spider, etc), do you get their spellbook? It's not listed as treasure…
The DM is free to add treasure, including spellbooks, to an adventure.
of course DMs can do what they want, but what is the intent? NPC wizards don't have spellbooks unless listed as treasure?
If we really want a spellbook to be found as treasure, we write it into an adventure. Otherwise, we lean on "Equipment" (MM, 11).
The section on equipment that Crawford mentions states:
A stat block rarely refers to equipment, other than armor or weapons used by a monster. A creature that customarily wears clothes, such as a humanoid, is assumed to be dressed appropriately.
You can equip monsters with additional gear and trinkets however you like, using the equipment chapter of the Player’s Handbook for inspiration, and you decide how much of a monster’s equipment is recoverable after the creature is slain and whether any of that equipment is still usable. A battered suit of armor made for a monster is rarely usable by someone else, for instance.
If a spellcasting monster needs material components to cast its spells, assume that it has the material components it needs to cast the spells in its stat block.
In this case, virtually all the spellcasters in the Lost Mine of Phandelver adventure use the evil mage statblock:
Spellcasting. The mage is a 4th-level spellcaster that uses Intelligence as its spellcasting ability (spell save DC 13; +5 to hit with spell attacks). The mage knows the following spells from the wizard’s spell list: [...]
(The final boss has a similar feature, and the only other spellcaster statblock used in LMOP is the flameskull, which are undead beings created from the remains of dead wizards.)
Though they use the wizard spell list, they do not necessarily need to follow the rules for PC wizards - NPCs are not built like PCs, and do not need to follow the same rules. Even if they were built similarly, however, even PC wizards only need their spellbook to change their list of prepared spells at the end of a long rest; thus, the enemy need not have such a spellbook on them in order to cast their spells.
The question "How to destroy a spellbook when the wizard dies?" discusses a case in which the querent doesn't want their player's wizard to have access to their NPC wizard's spell list.
As KorvinStarmast's answer points out, Volo's Guide to Monsters provides statblocks for a number of NPC archetypes that are similar to PC classes or subclasses, but as with the above case, the statblocks do not include spellbooks or mention that the NPC requires one.
(Other answers to that question also point out how such spellbooks may have been hidden or set to be destroyed, which could be given as in-universe explanations in case your players reject the out-of-universe explanation that PCs and NPCs are built differently.)