Is it possible for a syphilis chancre to show up 5 months after sex?

It is possible : Yes it is. It can take 6 weeks to develop signs and symptoms.

No, too long: Chancres generally appear 10-20 days after exposure, occasionally up to 6 weeks. Five months is too long. Herpes is the most common cause of otherwise unexplained genital ulcer. See a doctor or clinic soon; waiting until the lesions clears up will make accurate diagnosis much more difficult. Good luck!