What do you call a person who installs window panes?

The person is a glazier: one who sets glass

A glazier, sometimes spelled as glazer, is also known as a glass worker. The Free Dictionary mentions that a glassworker (one word), is a person who cuts flat glass to size.

Common synonyms include: glass cutter, glass-cutter; glass installer, and glass setter

Not to be confused with: glassblower, glassmith, or glassmaker, who are workers/ tradesmen/people who make glass.

One final synonym comes from the American website, Craigslist, which defines the job position of a Glass Technician as

Our client, […], is looking for a dependability enthusiastic candidate for an apprentice Flat Glass (Glazier) Technician for their growing team in St. John's, Newfoundland.

Responsibilities

• Repair/ replace commercial and residential glass and glazing systems
• Measure, fabricate and install mirrors, glass and custom showers
• Perform mobile service glazing

Consider window installer:

Should I Become a Window Installer?
Window installers fit pre-made windows into window openings and door frames of homes and buildings.

(study.com)

From service reviews of American Window Company:

The installer Manny and his assistant were quick, neat and very professional. My windows look awesome and I can already tell a difference in sound and insulation

The installers left the house as if they had never been there.