Or make the Pokemon from other regions, and like Alolan version Pokemon very rare or something?
While I generally agree with Starlight, I'd also like to emphasize her point about how Alola forms of old-gen Pokémon exist in Alola because they've
adapted to that region's environment. Thus, you don't really find Alolan Vulpix or Sandshrew or so forth wherever you find their "normal" variations, specifically because there's no reason for them to take on those adaptations just anywhere else.
However, if you
really want to use them, there's still probably a way to work it out—you'd just need to take into consideration the environment of your setting. (Let's assume that you're
considering setting your fic in Kanto but haven't really made a solid decision yet—or that mentioning Kanto was an example.) For Kanto, an Alolan Vulpix might not make sense because the region's temperate. Short of Seafoam Island (which is a watery cave and thus the wrong ecosystem for fox-like creatures anyway) and a few anime-exclusive locations, there aren't really that many places shown in Kanto where a Vulpix that's adapted to snow can exist. However! If you decide to set your fic in a fan region where there are snowy mountains or if you decide to set your fic in, say, Sinnoh (home of Mt. Coronet, a snow-capped mountain), then absolutely, you could totally have Alolan Vulpix. The point is, canon states Alolan forms are environment-dependent, but unless that actually means Alola has a super-unique/magical environment, you can still have the right factors to recreate those forms. You just need to keep in mind that not every region has the right combination of factors on-hand.
On a negative side, though, there's also the fact that we don't actually know a whole lot about Alola Pokémon in general so far, and that's also important to keep in mind. Sure, we know their names, types, and brief overviews of new moves and abilities, but we don't really know full movesets, move specifics (How powerful
is Revelation Dance, anyway? What type are half of these signature moves?), or in some cases, evolutions. So it'll probably be harder to include Alolan Pokémon because until you know specifics, you're probably not going to have some of the vital details needed to bring these Pokémon to life. (Like, well,
evolutions.) That and you'll need to be extra aware that you may have to "update" your Pokémon or your story as new information comes out. This is one of the less dodgy reasons why folks might tell you it's a good idea to hold off on using Pokémon from a newly announced gen if at all possible. (The slightly more iffy reason involves the fact that some people feel it's a bit gimmicky to use upcoming gen Pokémon, but that's really a matter of personal taste.)
Tl;dr, true, no one's really stopping you, and so long as you don't do it excessively (i.e., so long as you don't set your story in one canon region but then only ever show another gen/region's Pokémon for no apparent reason, effectively ignoring/replacing
all of the native Pokémon), you do you. But it's definitely a good idea to take into consideration the region's environment when choosing what Pokémon to show off, and it's
equally a good idea to be aware that Pokémon from as-of-yet unreleased gens require a bit more work to pull off due to the lack of information you'll have on-hand.
Good luck!