There's no question that vaccinations can be literally life-saving, and most horses sail right through the process without problems. But vaccine reactions do occur in horses.
The usual, and least serious, vaccine reactions in horses are similar to what you yourself may experience after a vaccination. Fatigue, low-grade fever and local soreness at the injection site are common in the first few days to a week or so afterward. Horses may also go off feed for a while. These are symptoms of the immune system responding to the vaccine. No specific treatment is usually required, although you may opt for a dose or two of phenylbutazone or flunixin (Banamine).
The next-most common is a more extensive reaction around the injection site, consisting of heat and swelling that extends over a wide area, and may even make it painful for the horse to move his neck. When these occur within the first day or two, they're usually exaggerated reactions to either the specific organism you're vaccinating against, or to the adjuvants in the vaccine, which are nonspecific immune system irritants/stimulants included to help boost the response to the vaccine. Your veterinarian should be consulted about treatments for this type of reaction, which may include bute or Banamine, hosing or corticosteroids.
When swellings at vaccination sites appear several days or more after a vaccination, the chance that the area has a bacterial infection/abscess is higher. These horses should always be seen by the veterinarian since some deep muscle infections could be fatal. Your vet will decide if antibiotics are warranted and will likely also tell you to use hot packs.