It can be normal for them to be a little funky as in sore or tired directly following vaccinations.
If the nasal discharge is thin and clear, I usually don’t get too alarmed.
If your horse reacts to vaccines, consider another injection site. I have a gelding that, true to his dramatic persona, gets a lot of swelling after any vaccination. The vet gives injections into his chest muscle, and he gets swelling between his front legs that subsides after a few days. I call it his “boob” :winkgrin: The swelling between his front legs doesn’t bother him at all and he can be exercised/ridden. If it happens on his neck, it really bothers him.
If there is hot and hard swelling, solid colored/yellow thick nasal discharge, and significant lethargy, then I’d consult your vet.