I was going to stock up on my favorite percussion caps. Do these things go bad with age? How long are they good for?

I don't believe they do. IMHO the biggest issue is moisture. keep them in a dry place and they should last a long damn time.

Keep dry and cool.

I have heard they do online, but it hasn't been my experience.

Doc

I bought a very old tin of winchester caps. What ever is inside them that makes them go bang was all dried up and on the bottom of the tin. I have no idea on the exact age or how they were stored all those years but yes they will eventually go bad.

The era percussion caps were made will most likely determine whether they go bad.
Since the early 1800s, when percussion caps were introduced, up to the late 1800s their explosive compound was mercury fulminate, often mixed with other compounds.
By the 1880s, potassium chlorate began to replace mercury fulminate.
Potassium chlorate was used well into the 20th century, but it's essentially a salt and attracts moisture. Potassium chlorate was used in primers for decades, and rusted many, many bores.
Today's percussion caps are likely a compound that doesn't attract moisture, and is considerably more stable than previous compounds.

I've fired percussion caps made by Winchester, Eley and Goldmark's that were manufactured in the 1870s or earlier. They all went BANG with varying authority. Seems like they'd all do the job.

In the 19th century, there was a dizzying array of sizes for percussion caps. They were not often numbered, but simply listed on the tin or in catalogs as Army and Navy Revolver" or Shotgun and Rifle."
Often the only difference was the length of the copper skirt. Short skirts were for pocket guns that had small, short nipples. Longer skirts were for the full-sized Colts, Remingtons, etc. Shotgun and rifle nipples came in all lengths, and so did their proper percussion cap.

For a lengthy, fascinating history of what explosive compound has been used in percussion caps and primers for reloading, go to:
https://www.bevfitchett.us/ballistics/priming-compounds-and-primers-introduction.html

I bought 5000 Remington #10 caps from:
https://www.midsouthshooterssupply.com/dept/muzzleloaders/shooting/caps-and-primers

Each 100 count tin is in the same plastic packaging as you'd see hanging in a store. I store them in my climate controlled house and doubt they will go bad before I fire them.