Yes, but it is only incidental that they may do so and it is not required to cast warlock spells known as a ritual.
Book of Ancient Secrets says:
You can now inscribe magical rituals in your Book of Shadows. Choose two 1st-level spells that have the ritual tag from any class’s spell list (the two needn’t be from the same list). The spells appear in the book and don’t count against the number of spells you know. With your Book of Shadows in hand, you can
cast the chosen spells as rituals. You can’t cast the spells except as rituals, unless you’ve learned them by some other means. You can also cast a warlock spell you know as a ritual if it has the ritual tag.
When you learn the invocation, you automatically choose two 1st-level spells that have the ritual tag from any class list. They do not count against spells you know (this is important for the next piece). These spells can only be cast as rituals unless you also know them through other means.
Finally, you can cast any warlock spell you know as a ritual as long as it has the tag, meaning that you need not inscribe your warlock spells in your little black book to cast them as rituals (though you still could, incidentally).
On your adventures, you can add other ritual spells to your Book of Shadows. When you find such a spell, you can add it to the book if the spell’s level is equal to or less than half your warlock level (rounded up) and if you can spare the time to transcribe the spell. For each level of the spell, the transcription process takes 2 hours and costs 50 gp for the rare inks needed to inscribe it.
Here, the invocation description is telling you outright that you can in fact add new spells to your book as long as they have the ritual tag. It says "when you find such a spell" but in this case "find" must be equivalent to "have access to" or "know" the spell, whether you've earned your own spellbook and inscribed the spell there (multiclass into wizard), learned the spell as a sorcerer, or even come across someone else's spellbook or a scroll that contains the spell. Under this interpretation, you could even record your warlock spells that have the ritual tag in your tome. Then, at level up when you can swap out warlock spells known, you can replace the spell recorded in your tome with something else, while still having it available as a ritual.
To interpret the wording of the invocation to mean that you can only inscribe spells you find, and only when you find them, seems pedantic and unnecessary. It creates a dissociated rule where there is a mechanic that has no apparent underlying association with the game world itself. That is, it becomes purely arbitrary and creates an in-character limitation for no apparent in-character reason. Why could a warlock inscribe into his tome a spell from a scroll he just stumbled across, but could not inscribe into his tome a spell he has known for weeks or months?