Yes, but only 1st-level spells
PurpleVermont has already argued why such a multiclass could only write down spells of a level they could cast as a Wizard (so actually up to 2nd level, if they take Wizard 3). I disagree with Vermont's argument that stolen spells could not be transcribed at all, however.
The rules don't clearly state that you have to have seen a spell written down to be able to transcribe it, that's a matter of interpretation--and I disagree with the interpretation:
Copying a 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.
The fact that different wizards use different notation implies that the written form of the spell is not an inherent part of the magic, merely an aide-memoire. And the requirement to practice the spell until you understand its V/S requirements -- precisely the thing which the Arcane Trickster can instantly intuit -- strongly suggests that this is the most important part of "learning" a spell, and the rest is merely paperwork that your first-level Wizard training would absolutely prepare you for.
The question has a dnd-5e tag, but in the interest of providing an avenue for the rather interesting concept, here is a lore answer from earlier editions. Most of what I write in the next three paragraphs come from an article named "The Cult of Ao: Who Watches Those who Watch over You?" by Steven E. Schend, published in Polyhedron magazine issue 94.
After the Time of Troubles (ToT), a cult was formed around the worship of the overdeity Ao. While the cult lost most of its followers once they realized that Ao did not grant any spells, some stayed. Those who stayed formulated tenets that reflect what Ao had declared during the ToT. In summary, as far as I understand, these tenets roughly state that Ao wants a balance, all the deities need to tend to their portfolios and to the needs of the worshippers in order to maintain that balance, the duties of each deity and his/her followers are equally important.
Thus the cult approaches all deities from the same perspective. If the worship of one deity is too strong in a given region, they support the opposing deities. They watch churches to see if the priests are really doing what they should be doing to further the portfolio of their deities or simply trying to amass only personal gains. For example they sink a ship full of opulent cargo meant to be used to furnish Sune's temple in Waterdeep, and that cargo ends up in the shrines of mermen deities.
Members of the cult were not faithless, they had their patron deities. For example, the leader of the cult at that time (circa late 1360s) was a wizard and a worshipper of Oghma. Some were even priests. They were usually very devout and embraced a conservative interpretation of their gods' teachings, as keeping the balance meant the clergy of each deity had the duty to act precisely in the way that advances the portfolio carried by that deity.
A lot has passed since the publication of the above 2e material, but remember that the people of Faerun have encountered Ao again very recently, as he announced the end of the Era of Upheaval, concluding the Second Sundering. Given that the 5e FR has the deities not interfering directly, the fate of the world is left to the hands of the mortals, and the cult might have rekindled. The divine order of FR needs better informed mortals now more than ever. Your character could be a devout cleric of Oghma or Savras, as you mentioned in your question. Other worthy candidates, who support the Knowledge domain and could support cleric members of the cult of Ao, are Deneir, and Jergal:
Deneir: a power who is working with Metatext, a work that records the whole of reality. He is also closely allied with Oghma. Your character might be one who understands the order dictated by Ao needs to be searched in every corner of the world, collected in writing and brought to the masses, who need to understand that the deities play important roles in that order and need to be worshipped.
Jergal: a power that used to hold Kelemvor's portfolio in the ancient times. He would be very familiar with the faithless as he is the scribe that keeps the records of the dead. Warning: His uncaring ethos seeing life as a temporary step in the eternity of death requires careful consideration.
Best Answer
I'm not sure I'd call him "active" but Ioulaum would know how to make one, being the Archmage that invented the Mythallar.
Last I read he had transferred his consciousness into an undead elder brain and he may have a steep price for giving out this information.