In general – crafting requires a formula, but not a formula book
Let’s start more generally than the alchemist. Crafting an item requires that you have the item’s formula (Core Rulebook, pg. 244):
Craft: … To Craft an item, you must meet the following requirements: …
You have the formula for the item;
You don’t have to have it in a formula book or any particular form. You merely need the formula in some fashion. That being said, formulas are typically in a written form (Core Rulebook, pg. 293):
A purchased formula is typically a schematic on rolled-up parchment of light Bulk.
The description of the item formula book on page 290 clarifies that no formula book is required to craft something. It merely holds formulas:
… there’s no need for you to copy (formulas) into a specific book as long as you can keep them on hand to reference them.
This is the core answer to your question: in general, crafting requires that the crafter have an item’s formula. It need not be in a formula book, but it must be “on hand to reference”.
Some alchemist features require a formula book.
There are two reasons the alchemist is worth mentioning specifically: it has a class feature called formula book, and it has other class features which reference formulas.
The formula book class feature changes nothing about how to craft items. It only provides a bunch of formulas at level 1 and more as you level up.
The Advanced Alchemy class feature allows alchemists to create items as part of the morning preparations. Because of your comparison to wizards, I believe this is the feature you are referring to. Advanced Alchemy does not require a Craft roll or Crafting task, and therefore skips the previously mentioned requirements for crafting. However, it explicitly requires that a formula be in your formula book, not merely available (pg. 72):
… choose an alchemical item of your advanced alchemy level or lower that’s in your formula book, and make a batch of two of that item.
There is no mention of having to consult the formula book while creating these items, but the formula must be in there.
Quick Alchemy is different, in that it’s requirements explicitly mention that you must have the formula for the alchemical item available. It’s not enough to have the formula written in your book, you must have it present.