Let me start by stating two points that are relevant to this answer. Feel free to skip this section if you want to jump directly to the 5e material.
In previous editions, official FR material dictated that paladins receive their powers from deities. In general, deities are a major part of this polytheistic setting, and mortals and powers are well-aware of this. Deities get to run the (meta)physics of the world and not having faith diminishes their power and causes imbalance. Our real-world secular/religious sensitivities are not attuned to this way of thinking, perhaps a good analogy is having your children vaccinated. When most people have vaccination, we get a herd immunity; likewise when most people have faith, the FR world 'runs'. As such not just paladins, but all people are expected to have a patron deity. If not, they are "disciplined" in the afterlife and become part of the "Wall of the Faithless". The attempts by the god of the dead, Kelemvor, to resolve this seemingly unfair treatment of the faithless has been depicted to have serious consequences in novels, which resulted in a change of his personality.
In spite of the central role faith plays in the FR of Greenwood, you are more than welcome to reject it; or find ways around it (like a faithless paladin who is still being watched over by a deity like Torm). Throughout its whole history, Ed Greenwood and the various versions of any written material have always highlighted the fact that each FR campaign belongs to the individual DMs. So feel free to rule however you like.
As for the officially printed material you have asked for, below is what is available.
What do the 5e Sources Say?
Sword Coast Adventurer's Guide states (page 131):
When ... a warrior also has great devotion to a particular deity, that god can reward the faithful with a measure of divine power, making that person
And yet on page 132:
Most paladins in the Forgotten Realms, like clerics, are devoted to a particular deity.
These two sentences might seem to be in conflict, unless we interpret them to state that 5e FR paladins get support from one or more deities. This interpretation is supported by lore from earlier editions (see for example the Triadic Knights) as well as the text on clerics (page 125):
A typical cleric in Faerun serves a single divine patron, but some individuals feel called to serve a group, ...
D&D Adventurer's League
It is worth noting that the D&D Adventurer's League FAQ has switched its position on this issue over time. (The change might be correlated with the introduction of non-FR campaigns to DDAL.) In version 2.5 it stated:
Does My Paladin Have to Worship a Deity? Yes, though your character’s alignment isn’t required to match that of their deity.
While version 3 states:
No. What makes you think you do? Dunno what you’re talking about.
WotC Suggestions for Content Creators
Finally, Wizards of the Coast has a set of style guides for their official play writers and have been made available for all content creators on DM's Guild. Note that these guides are suggestions and not rules, but for the sake of completeness, I would like to quote the relevant parts. D&D IP Guide states:
Divine magic is practiced by the most devoted followers of D&D’s gods: clerics and paladins. ... Paladins are the holy warriors of D&D. Like clerics, they serve a deity and channel divine power to achieve their goals (or the goals of the temple or holy order they serve).
Forgotten Realms Style Guide confirms:
Paladins are warriors of unusual virtue and piety rewarded by a good god with divine power. ... paladins are typically judged more swiftly and harshly by their gods.