"Running royalties" are unlikely (no copyright), but there are ways for authors to make money
While the nice answer by Theik provides information about modelling making money as a downtime activity in 5e (and I agree with his point), it really does not answer the question of "collecting royalties as an author" in the Forgotten Realms.
The answer is "unlikely" if you are considering running royalties in which you will keep on getting paid as your work is kept on being published, at least in the most metropolitan city of Faerun, Waterdeep. You can read about printing presses, bookshops, and famous publications in the 4-part series Small Presses of Waterdeep by Ed Greenwood published in 2003.
Anyone can copy any book without legal penalty in Waterdeep, and printers amass libraries of chapbooks printed by their rivals so that they can plunder for ornaments and illustrations when a "new" book must be swiftly assembled.
Please note that I am not saying that your character cannot make money, it is just that you cannot write a creative work at some point and expect to keep on making money out of it in perpetuity. For example, here is a part from a flood of unofficials tweet by Ed Greenwood about how famous authors like Volo make money:
Bookshops became fixtures of the Sword Coast port cities and all major Heartland trading cities and ports by 1475 DR, and places like Waterdeep, Silverymoon, Derlusk, Baldur’s Gate, and Suzail had local bestsellers and a marketplace of “here’s what’s coming” and “read a chapbook excerpt from the forthcoming new sequel to X by talented and famed Author Y” by 1478 DR. Traveling merchants (and simple peddlers, going from hamlet to village) since then have aided in spreading this ‘culture’ everywhere. So Volo is signing copies of his latest as just one author among many (albeit a notorious one who can claim a long and successful career), by the 1490s.
That is why modelling the attempt as a collective downtime activity (finding sponsors, authoring commissioned work, printing, giving lectures, book signing, etc.) is a very reasonable game mechanic.