Hello, my name is***** and I have over 20 years of experience as a veterinarian.
I am sorry to hear that Snuggles is coughing, lethargic and eating and drinking poorly.
When we see them react to vaccines it is usually within the first 48-72 hours. Because it has been 5 days a reaction to his vaccines seems unlikely.
It is very likely that he has picked up an upper respiratory infection and being a young kitten his immune system is weaker than an adult and likely didn't have time to form protective immunity from his vaccine by the time this virus attacked. An upper respiratory infection in cats is just like a cold in you and I, and these are commonly caused by a virus, the most common one being Herpes virus. These are easily caught by breathing in virus particles in the air from a sneeze or nasal or eye discharge. Some cats get sicker than others and young, healthy adult cats seem to tolerate them and fight them off better than very young kittens or older cats. I am glad he is still eating and drinking some, but his lethargy does concern me, and warrants your watching him very closely.
You can help him feel better by adding warm water to his food to make it smell more (they don't eat if they cannot smell) as well as making it easier to swallow. Also the more fluids he gets the better. Offer tuna juice, low salt chicken broth, run the tap if he likes to drink out of the sink.
Take him into the bathroom with you if you run a hot bath or shower as the steam will soothe his sore throat and airways. If his nose become very congested you can use sterile saline to loosen the thick mucous and remove it. He won't like it but it will help him breathe and be able to smell his food better. You can also use sterile saline to remove eye mucous if it accumulates.
You can give him an amino acid supplement called L-lysine at a dose of 500mg orally twice daily. If this infection is due to Herpes this amino acid interferes with virus replication and will shorten the infection's duration and severity. Good supplements to try are made by the Viralys brand which comes in a powder to add to the food or a tasty gel.
Some lethargy is understandable, let him rest as he needs rest to get better, but if it becomes extreme we need to worry he is developing pneumonia. If he is normally allowed outside I would keep him inside until he is back to his normal playful self.
If he runs a high fever (more than 104F), has a green or yellow nasal or eye discharge, stops eating even with coaxing and clearing his nose and eyes, or starts coughing such that he cannot get comfortable, or having constant difficulty breathing then he needs a veterinary exam. Sometimes these upper airway infections turn into pneumonia so that's what we need to guard against.
In most cases antibiotics aren't needed and can contribute to a decrease in appetite so I don't tend to prescribe them unless I feel there is evidence of a secondary bacterial component. These include a green or yellow eye or nasal discharge, evidence of pneumonia upon listening to their lungs or an infection that lingers beyond the normal 7 to 10 days.
I highly recommend testing him for feline leukemia and feline immunodeficiency viruses if his respiratory infection lingers. These immunosuppressive viruses will make a simple infection much worse as they stop the immune system from fighting infections the way it was designed to do.
Please let me know if you have any further questions.