Tjyylli

So I got a fantasy setting that left the Medieval era and entered the Renaissance with strong states that can maintain both order and infrastructure across their territories. While there are 'wild areas' where the rule of law is relatively weak, you can bet, if someone tries to set themselves up as a bandit king and tries to control an area, soldiers are going to come knocking at their door within months at the latest.

Now I want to put goblins into this setting in realistic places. To define goblins: in the case we are talking about creatures approximately the size of a five year old with very good night vision, relative quick reflexes and poor strength. They breed and grow rapidly, with a goblin being an adult at one year in age and female goblins birthing somewhere between four to ten children at a time after a six months long pregnancy. Culturally they are fairly primitive and tend to struggle with higher technology. They are more likely to steal and scavenge than build or create, although they tend to have a good grasp of teamwork and ambush tactics due to a strong family based culture.

Narratively I'm planning to use them for horror, and I'm fine making a few tweaks, as long as they are still very much flesh and blood creatures. So what kind of places could they turn up in?

They would either be driven to a few places as far removed from civilization as possible, or become part of this new civilization like rats became a part of cities.

Wild Goblins

The humans would simply wipe out any goblin nest they find that's in their way. That leaves only places like thick woods, mountains, swamps or other waistlands that have no value for humans and are therefore traveled very infrequently. Due to their night vision, the goblins would hide during the day and scavange at night.

City Goblins

If the goblins are civil enough to be tolerated in the streets of a city (like stray dogs and cats are tolerated), they might find a comfortable life among the sewers and garbage dumps of humans. The humans would quickly notice that they can make the goblins work for them for laughably low costs. Undesirable jobs like sewer maintenance, waste collection and especially dangerous jobs could safeguard the life of goblins among men (at least those goblins who survive their jobs ... You have to keep their numbers in check somehow after all).

Depending on their intelligence, they could either be in the wilds, forever running and hiding, or be enslaved to humans.

In the case of the former, adding to their night vision capability, they would adapt to be nocturnal creatures. Hunting, scavenging, and gathering resources at night when the world is asleep and resting while hiding from danger lurking in the daylight in somewhat safe, secluded areas such as caves, dense forests, or even abandoned civilizations.

For the latter, however, the oh-so-poor goblins would be sold accordingly to their qualities and traits at auctions or slave shops for a very low price. Their lifespans would be shorter than it would be if they weren't worked to death by humans. Using goblins as workers would be something common and a cheap price to pay to get something done without precise thinking and complicated handiwork. So, they would probably live in slave shops, workers' cottages, or simply traveling while being dragged around by slave traders.

Caves: Their night vision allows them to see in the caves. Their small stuature makes maneuvering through caves easy. Their quick reflexes help them catch the uncommon morsels (insects, fish, bats) that dart throughout the cave. However, any small dark area would do (underneath buildings etc.). This would allow them to hide from humans and grow in number.

As others have stated, they can live underground and be enslaved by other races, such as they are in many story universes. Low intelligence, low strength, they pretty much have to live secluded from everyone in a basic undetectable location -- underground. They don't have magic to hide behind like some Elven races, don't have strength to be able to bring items for trade and not get steamrolled by another race. Arguably they won't have much of a military or have very basic military tactics (ambush, like you mentioned). Could be nomadic to stay distant from various Dwarven races, or be cannon fodder for them (like they are for the Drow in Forgotten Realms).

Caves can be entrances/exits to their underground cities where they come above to get resources. Food sources could be rats, spiders, bats, underground lakes with fish. They could also tunnel like moles (or even use moles TO tunnel) and pop out at different locations, such as under castle walls, to snatch supplies or hostages, or use ambush tactics similar to Vietnam during that war.