Asbestos is another one of those resource in Fallout 76 where you THINK you will always have enough, but as soon as you start crafting weapons and armor, you run out quite quickly.
Here are a couple items where you can most easily grind for Asbestos by breaking them down:
Biometric Scanners
Catch the Commie Board Game
Chalk
Cigaratte Cartons
Extinguishers
Coffee Pots
Oven Mitts
Rat Poison
Teapots
The problem with find these items in particular is that they seem to be EVERYWHERE… but also no where when you need them. They typically can be found in office buildings and homes, the rat poison moreso in garages and basements of buildings.
Another quick note to farming Asbestos is to know that Fog Crawlers actually give a good amount when you kill them, so make sure to hunt them down when you can.
A few locations to find these items lying around would be: