Don't know whether to report this as a bug or a task, but anvils in Minecraft: Java Edition are very clearly waterlogged while falling inside a body of water, leading the player to believe they are waterloggable in general. However, as soon as the anvil touches the ground / becomes stationary it is no longer waterlogged and there is no way to make it so. This, of course, is just a visual problem and does not affect your interaction with the anvil in any way. However, it does make an annoying problem for builders such as myself who might want to use it in underwater construction. Since most other blocks in the game (Trapdoors, stairs, walls, etc.) are waterloggable I see no reason why anvils should'nt be - and the game clearly shows that there is a way to do it. 

Below I have attached two screenshots: "Screenshot 1" shows the anvil block in the context of a build. The anvil is stationary and therefore not waterloggable. "Screenshot 2" shows an anvil falling inside a body of water before touching the ground, where the anvil is clearly waterlogged.

(For some reason the website is not letting me select a Minecraft version below 1.16.1, but this issue affects all current minecraft versions and Snapshots.)