I have a 19 yr old gelding that has gotten a similar reaction to his 4-way shot the past several years.
The spot it was given (neck muscle) gets a little puffy and he doesn't want to put his head down to ground level to eat. It goes away eventually but he's noticeably sore for 3-4 days.
The vet said it's due to the tetanus part of the vaccine. He tried giving the shot in two different sides of his neck last year and my guy still got a reaction. I think next time I will see if we can give it in the hindquarters. At least then he can eat comfortably. Nothing much sadder than a horse that wants to eat grass out on the trail and can't put his head down far enough to reach it.