I just caught a Vulpix, which I hadn't seen before. If I go back to the same area is there a chance I'll see another one? There wasn't anything special about the area that it was in, just a residential sidewalk
While the exact spot will not spawn the exact same Pokemon repeatedly, the area you are in definitely affects the Pokemon which appear.
For example, my friend who lives on the beach sees water Pokemon all day, while I see more Rhyhorn and Eevee.
From the FAQ:
If you don’t see any Pokémon nearby, take a walk! Pokémon love places like parks, so try visiting a local recreational area. [...]
Some wild Pokémon appear only in certain environments and climates. For example, some Pokémon may appear only near lakes, oceans, or other bodies of water.
Also see this related question.
Common, uncommon pokemon always appear in the same general area, if not spot. However I have experienced certain buildings which are on google maps but not on the pokemon go map, that spawn new pokemon right on top of it consistently. Think of a little snack bar in a park, the snack bar always has some pokemon when I go to it.
Other than that, I've seen a pattern like this:
There are 4 different locations I go to.
a,b,c,d spawn in all locations.
e,f,g spawn in location 1 & 2.
h,i,j,k spawn in locations 2&3.
l,m,n spawn in 4.
o,p spawn in 1.
So you have some pokemon only spawning in certain areas and some pokemon are more commonly seen everywhere.
So according to this insight when I want to get more Eevee candies, I go to the place I've caught a few Eevee's before. I am positive it doesn't spawn in other locations I go to at all.
You can also guess what a place spawns by looking at a couple of gyms, if they all have Hypnos, than its probably a good place to catch Drowzees.