One thing to do, if you aren't already, would be to use spellcheck. It should catch places where you miss a space, like "andcheck" in Chapter One and "todo" in Chapter Two. Most writing programs will also allow you to add words to their internal dictionary, and pasting in a list of pokémon names will help you catch cases where they're mispelled. I spotted errors in spelling "Solgaleo," "copperajah," "rhydon," and "skiddo" as I was reading along.
You don't capitalize the first word after a colon (or a semicolon, for that matter). So here:
The two scientists at the door immediately straighten up and give him the salute of Team Rocket: One fist against their heart.
It should be "the salute of Team Rocket: one fist against their heart." (That was a fun detail, though!) Likewise here:
Now here’s your deal: You walk out of this base without a target on your back, but you report to me.
This should be "Now here's your deal: you walk out..."
Cherry and I were in silent agreement: We didn’t want to risk it.
And this should be "Cherry and I were in silent agreement: we didn’t want to risk it."
On a similar note, I'm not totally sure what's going on with your dashes. Style guides can differ a bit with those, but generally you don't capitalize on either side of them. But here we run into these sentences, not far apart in the narrative:
The pidgeot flew past us, the rider looking back at us as it went—He was looking at us. [...] There were a pair of bulky goggles strapped to his face – was that what he was using to see us?
The first sentence uses an em dash, no spaces, and capitalization. Which I think is fine except for the capitalization. Then in the second sentence there's what I
think is an en dash with spaces and no caps, which I think is fine except for the fact that it's an en dash. One way or another, I think you want to pick a style and stick to it throughout the story.
Final broad category: when you use a comma with a conjunction (comma plus "and," comma plus "but," etc.), there should be a complete sentence to each side. If one side isn't a complete sentence, you don't want a comma. So here:
Arceus deftly laid hoof upon the ground, and bent over until he could reach the crouching espeon’s level.
"Bent over until he could reach the crouching espeon's level" isn't a sentence, so there should be no comma. Likewise here:
Another stamp of Arceus’ hooves, and bright pink power swirled out of his chest.
"Another stamp of Arceus' hooves" isn't a sentence, so no comma in this sentence.
He pulls out a silver, slightly rusted pocketwatch, and keeps the time with it.
"Keeps the time with it" isn't a sentence, so no comma before "and."
One flash of light later, and we were on the street outside the house.
"One flash of light later" isn't a sentence, so no comma in this sentence. And so on; there were further sentences with this problem.
Wrapping up with some one-off notes...
Chapter One
Thunder rumbles through the distance, low and mighty through the rough, churning sea.
The repetition of "through" here feels off, and thunder isn't usually said to rumble
through the sea.
The helicopter comes to a swift landing at the bank of a vast cliff.
cliffbank?
He sports long, wavy hear, a goatee, and looks like he’s been existing purely on coffee for the last twelve days.
wavy *hair, not "wavy hear"
“Y-you didn’t have to come all this way. There’s a storm out, you know?” laughing nervously, he gestures to one of the windows, where the furious rain can be seen but barely heard.
"Laughing nervously" and the rest of that sentence don't have anything to do with the preceding dialogue--the dialogue could be omitted and it would still make sense. You therefore want "lauging" to be capitalized; it's not a speech tag.
An expression of interest has broken his normally cold face.
"Broken his face" sounds painful, heh. Maybe you meant "broken over" his face?
Chapter Two
Now I actively wish I didn’t dream about it, because teleporting feels like flushing yourself down a toilet. Way to flush my dreams, property of transference of energy.
"Property of transference of energy" struck me as kind of clunky here; maybe something like "way to flush my dreams, laws of physics?" I thought this was a cute joke and a fun way to describe teleporting, though.
I pulled out from a shelf a glossy orange folder for Victini, which I should really keep more around of, and held out my free arm.
"Which I should really keep more around of" reads pretty awkward to me. It's kind of tough to get that side thought into the sentence gracefully, I think; best option I'm coming up with is something like doing an em dash: "orange folder for Victini--I really ought to keep more of those around--and held out my free arm.
My first breath of the ambient dust and must from the hotel room hit me like a hyper rydon covered in stun spores.
Here's the "rhydon" misspelling. The simile isn't helping me understand things better, unfortunately. A "hyper" rhydon, so one more active than normal, but covered in stun spore, which I'd expect to slow it down and neutralize the hyperness. Not sure what you're trying to tell me with this one.
“Aww, seriously?” he complained, thrusting his arms.
Thrusting his arms where?
There were eight seats on the High Council in total, and I had prim and prepare each one of them.
I think you want a "to" after "had," and also you probably mean "primp" rather than "prim."
It’s mean, but every time I look at him, I imagine an absol that got half a fence stuck to its midsection, and had it encrusted with gems to help stave the embarrassment.
*stave off, and this is also one of those cases where you don't want a comma before the "and." Fence llama jokes are always good in my book, though.
I floated to the side with a wry frown, a notepad and pen I’d teleported in floating behind me.
I'm confused--I thought Ann flew back to her desk manually. So why would she have teleported anything in?
I suddenly tensed up, my psychic hold around the pen I tightening to the point where it nearly cracked.
Missing something after "I."
A car drove past us, stealing the voice outside of Cherry’s mouth.
I think I get what you mean, but "the voice outside of Cherry's mouth" sounds really odd to me. Maybe stealing her words, stealing the sound of her voice, something like that.
As long as I was around to clock in and off, Arceus wasn’t studying me close enough to notice.
It's usually expressed as "clock out" rather than "clock off."
Cherry raised an eyebrow, eyes still folded.
Eyes folded? I'm not sure what you mean by that.
“Just throw Uxie a snorlax-sized candy bar,” Cherry shrugged.
Shrugged is not a speech tag; it doesn't describe the way in which the dialogue's being said. Therefore you want a period rather than a comma after "bar."
Half the restaurant had to be looking at as weird at that point, but neither of us really cared.
at *us weird
I quickly shifted back into Mew once my brain what was happening, catching myself midair and halting to a float.
Once her brain what?
The move knocked Cherry off-kelter, and she dropped me.
*off-kilter
A black figure rode into the distant moonlight, and I saw that the pidgeot rider was banking around for another go.
This sentence makes it sound like there are two riders to me--one who just emerged into the moonlight and the one on the pidgeot. Using a pronoun intsead of an epithet in the second half of the sentence would fix that.
She swerved around a tight corner, knocking a streetlamp out of shape as we richoceted past.
*ricocheted
“They were able to see us. ” she repeated it again, to herself.
Random extra space before the quotation marks, and "she" should be capitalized.
I teleported into the gunshot seat.
Shotgun, not gunshot, though it might read better to use "passenger" seat.
No sooner had we looked in the direction of the city did we see the shape of the Alakazam standing all by itself in the parking lot.
The tense is a bit strange in this sentence; rather than "did we see," I think you want "than we saw."
Chapter Three
The hall was a proud palace, with a roof that lead up into the very peak of the mountain, where no normal human or pokemon could go for want of air.
*led, not "lead." When "lead's" pronounced like "led," it's referring to the metal.
Between them, a desk, and a door under the middle of the staircase that lead into a room unknown and hidden by shadows.
*led
*reinforced. "Re-enforced" would be enforced twice!
“Got a good feel for the place?” the voice came not from the man who had interrogated me, but from one of the two rocket grunts who had escorted me out here.
"The voice came not..." is a complete sentence and not a speech tag, so "the" should be capitalized.
I began to go through the list of major and minor things that had shown up in the paperwork today, focusing on the end on Zygarde’s departure
Not totally sure what's going on here. Maybe you meant "in the end" rather than "on the end?"