Where are the label templates stored in Word?

I second Charles's recommendation, for a couple of reasons.

First, the files that Avery offers as "templates" are documents (*.docx), not templates (*.dotx) as they're understood in Word's system.

As a result of the mislabeling of Avery's files, they don't appear in Word's list of (real) templates, which by default are stored in your Templates folder (%appdata%\Microsoft\Templates) or whatever folder you specify at File > Options > Save > "Default personal templates location".

When you download a file from a website such as Avery's, unless you get a Save As dialog in your browser to choose the destination, the file will usually be saved in your Downloads folder. Have looked there for your file? Failing that, use your browser's Downloads page (Ctrl+J in most browsers) to look for the file and its saved path.

There is an often misunderstood fact about the Label Options dialog that Charles mentioned. After you choose Avery US Letter in the Label Vendors dropdown, the product numbers appear in the list box. But they're not quite in numerical order (in fact, they're in alphabetic order): for example, all the numbers that start with 4 appear before any of the ones that start with 5, even though the last of the 4-starting numbers have five digits and the first of the 5-starting numbers have only four digits.

Do not use the Avery templates.

Instead use the Mailings Tab, Labels, Options button and choose your label.