@BrittanyC How are things going? Please keep posting thru the night...just a sentence or two to let us know how you and Oliver are doing.
I think I would plan on being at the vet's office when it is due to open...don't even call for an appointment, they can walk you in as an emergency. Go armed with a list of questions: What is his blood sugar now? What is his fructosamine? What is his ketone level? What is his temperature? What do you think has caused him to stop eating and why do you think he was/is vomiting? Is he dehydrated? Does the vet think he has an infection? Ask for a full chemistry panel and a CBC at the minimum...he might need to have other blood work such as his thyroid levels checked, is he feline leukemia negative? etc. He might need a urine specimen to check for infection, blood, etc? Maybe a fecal specimen to check for worms? He should get a good physical exam too...including: checking his retinas (can see the blood vessels), checking his teeth/dental condition and oral mucosa, checking his ears, listening to his chest and belly, deep palpating of his abdominal organs and his thyroid, checking for pain with full movement of each leg, checking for pain along his backbone, checking his anal glands.
It's a lot to remember, so take notes of the vet's answers, ask for copies of lab work and X-ray reports. Then if the vet still thinks Oliver is diabetic, ask for specifics of his insulin dosing. Or, what is the working diagnosis and how are you going to care for Oliver? When does the vet want to see Oliver again?
Then the very last thing you want answered is how you are to manage emergencies again? Where is the emergency vet, what is their phone number, etc.