This is "The Plague" by Teddy Keller (apparently the only SF he wrote), published in Analog in 1961 and collected in various Analog anthologies. It's available at the Gutenberg Project here.
Andy glanced at the lab report and his smile was as relieved as it was weary. "Our problem," he said, "was in figuring out what a writer does that a doctor doesn't—why girls from small offices were sick—and why senators and postal workers weren't—why college students caught the bug and people in a Tennessee community didn't.
"The lab report isn't complete. They haven't had time to isolate the poison and prescribe medication. But"—he held up a four-cent stamp—"here's the villain, gentlemen."
The big brass stood stunned and shocked. Mouths flapped open and eyes bugged at Andy, at the stamp.
Bettijean said, "Sure. College kids and engaged girls and new parents and especially writers and artists and poets—they'd all lick lots of stamps. Professional men have secretaries. Big offices have postage-meter machines. And government offices have free franking. And"—she threw her arms around the sergeant's neck—"Andy, you're wonderful."
"But there's no evidence it was a plot yet. Could be pure accident—some chemical in the stickum spoiled. Do they keep the stickum in barrels? Find out who had access. And ... oh, the phone call. That was the lab. The antidote's simple and the cure should be quick. They can phone or broadcast the medical information to doctors. The man on the phone said they could start emptying hospitals in six hours. And maybe we should release some propaganda. "United States whips mystery virus," or something like that. And we could send the Kremlin a stamp collection and.... Aw, you take it, sir. I'm pooped."
The general wheeled to fire a salvo of commands. Officers poured into the corridor. Only the brigadier remained, a puzzled frown crinkling his granite brow.
"But you said that postal workers weren't getting sick."
Andy chucked. "That's right. Did you ever see a post office clerk lick a stamp? They always use a sponge."