<<Disclaimer: Verify this information before applying it to your situation.>>

Everyone who responded said white pepper was GF.
One individual replied with information about Sysco/Alliant:

I was at g.f. camp in Minnesota the past few summers and they used the
Sysco/Alliant spices.  Most were g.f., especially the single spices/herbs.
The one exception to the rule was the Italian seasoning blend which did
have something that wasn't g.f. but I can't remember quite what it was.

My husband works for US Foods and he said all of the US Foods spices are
McCormick spices.  My understading is that McCormick single spices/herbs
are GF.  The blends may not be GF.

* Please remember some posters may be WHEAT-FREE, but not GLUTEN-FREE *