It is to my knowledge that bolt-action rifles with internal magazines originated during the 1880 with rifles such as the Lebel Model 1888 and the Lee-Metford. During the Late 1880s and early 1890s, many militaries began to transition away from singe shot weapons towards modern bolt action rifles, with the US adopting the Krag-Jorgensen in 1892.
Videogames such as Hunt: Showdown and Red Dead Redemption 2 portray such rifles as being available in the American west during the 1890s, being set in 1895 and 1899 respectively. In these games, the majority of characters depicted use western lever-action or single-shot rifles, however bolt-action rifles are still present. With how early these games are set, I doubt that these rifles would have proliferated as far into the West as they are portrayed as, especially since the majority of these rifles are used by the military.
With that in mind, I was wondering how common bolt action rifles would have been for your average citizen during this time period, especially since this is the era when these weapons started to become popular.
Probably not yet that common.
There was a small arms race in the 1800's, that started with the Prussian Dreyse "Needle-gun" in the 1840's. In the 1850's many European countries adopted various single-shot breechloaders. The advent of the 1866 Winchester pushed the further adoption of magazine weapons, like the bolt-action Swiss Vetterli . The US military lagged behind, not completely switching over to a breechloader until after the Civil War, passing over magazine weapons until the Krag Jorgenson.
However, the big change in 1888 was the French development of a workable smokeless powder. For large caliber military cartridges, instead of breech pressures of around 14,000 psi, smokeless powder pressures were around 40,000 and higher. That was too high for wrought iron and soft steel. Bolt actions were somewhat easier to design for those pressures, and so in the 1890's there was a general shift towards bolt-action smokeless powder rifles in most militaries. (The bolt action had other advantages, too- but that's been discussed here before).
But smokeless powder was not available for reloading in the 1890's- it was only available in loaded ammunition. For many in the rural areas, that was a problem, because they were used to re-loading, and anyone who wanted to reload had to stick to black powder. That meant there was really a smaller market for smokeless guns. Gunsmith Otto Niedner was making smokeless-powder rifles for early-adopters like Charles Whelan, but was frustrated with having to buy factory ammunition. Remington would alter their enormously popular Rolling Block design in 1897 to take smokeless powder ammo. John Browning would design the very strong 1895 Winchester to do the same. But neither company would produce a large-bore smokeless bolt-action until much later.
The only possible exception to this might be the huge numbers of black-powder Vetterli rifles that were dumped in the US and other foreign markets when the Swiss upgraded their arsenals to the Schmidt-Rubin in the 1890's. Much of the Vetterli ammunition was rimfire and could not be reloaded, but huge amounts of it were imported and sold cheap as well. While I know the Vetterli would be very commonly found circa 1910 in the backwoods of almost any US state, it's possible some were imported even earlier.
The Vetterli has quite a steam-punk appearance, and you'd think plenty of videogame designers would like it.