Depends on who collects the "soul". Try Gwaeron Windstrom for flavor.
First of all, I believe the following answer already covers your question: Where does an unaligned creature's soul go after death? [Do no focus on the title of the question.] Quoting the relevant parts:
In short, where the souls go, whether they were believers or not, is not an issue of alignment in the Forgotten Realms, it is about which deity collects them.
If you decide that the dead animal had a soul, it is likely be collected by the agents of Silvanus if it was a wild animal, or by the agents of Mielikki if it had been a companion of a ranger or other follower of Mielikki, or by the agents of Chauntea if it was domesticated. A particularly vicious animal's soul might also end up in Malar's domain, some deadly spiders might even go to Lolth's domain, as the DM sees fit.
I could add Gwaeron Windstrom to that list; particularly since he is worshipped by the rangers of the North and because his status as the Master Tracker would fit embracing dogs, animals who contribute to hunting and tracking.
Extra Lore: Beast Cults in the FR
Since this is a question tagged "lore" and "forgotten-realms", I will note down some pieces of ancient lore for the sake of completeness.
The very first published appearance of the Forgotten Realms deities is in Dragon magazine issue 54 from October 1981. In an article titled Down-to-earth divinity: One DM’s design for a mixed & matched mythos, Ed Greenwood introduces what would become the FR pantheon. On page 9, there is a table for "Cults of the Beast" which include a number of demi and lesser gods, only few of which would appear in future FR products (Aslan in the form of Nobanion, Lolth and Lurue). This table lists Roofdrak, Master of all Dogs.
Roofdrak itself is from the first and second printings of AD&D 1e sourcebook Deities & Demigods. It is a demigod from Melnibonean Mythos, created by fantasy author Michael Moorcock. Within Melnibonean Mythos, DD states (page 86):
It seems that every species is aware of the need for some sort of deity; therefore in the cases of dogs (for example), all dogs worship their own image, which in turn becomes Roofdrak, "Master of all Dogs".
Despite an agreement with the company Chaosium that would allow TSR to keep on publishing without copyright issues, Melnibonean Mythos would be dropped from further products. So we do not see any further mention of many of the members of the "Cults of the Beast" by name in the FR sourcebooks.
Yet, the flavor is still there. In the AD&D 1e FR campaign setting, we read about Beast Cults:
Similar to and more varied than the sects of the Elemental Lords, the Beast Cults are wild faiths, attracting the worship of savages, nomads, evil humanoids, and those removed from the normal circles of the world.
And later in the 2e sourcebook Forgotten Realms Adventures (page 36):
The beast cults believe in the existence of perfect forms of animals. A dog cult, for example, believes there is a perfect dog, who is leader of all other dogs.
A prominent feature of FR is that it is very much your setting. Feel free to concoct a story from all these pieces of lore as you fit.