NEWTOWN, Conn.—The number of people seeking permits to buy guns has surged in this town following the December massacre of schoolchildren by a local man, even as the parents of some victims had urged stricter weapons laws nationwide.

Through July 24, more than 200 people in Newtown have received new local pistol permits, according to a review of local records, surpassing the 171 new permits issued for all of last year. Such permits are prerequisites for Connecticut permits that allow people to purchase and carry pistols as...