Just ran into this problem when updating to the new input system. It looks like the fix is in 2020.2, why not have it in an updated 2020.1.x?

Had this issue as well. Using Unity 2020.1, Input System 1.0.0

Found a temporal solution here: https://forum.unity.com/threads/textmeshpro-input-field-new-input-system-webgl-no-work.926396/

Edit -> Project settings -> Player -> WebGL Settings -> Other Setting -> Active Input Handling and setting that option to "Both" instead of "Input System Package (New)

Same issue here.
Using Unity 2020.1, Input System version 1.0.0 and TextMesh Pro 3.0.1.
Tested with scenes that had nothing but a TextMeshPro Input Field and the normal Input Field. Neither worked.

I am having the same issue where characters are not displaying in the input fields for my WebGL builds. Both the standard and Text Mesh Pro input fields are affected, but they work fine in the editor.

Versions:
Unity 2020.1.2f1
Input System 1.0.0
Text Mesh Pro 3.0.1

I just installed the latest Unity 2019 and made sure all the packages were up to date and the problem still exists.

Versions:
Unity 2019.4.6f1
"com.unity.textmeshpro": "2.1.1",
"com.unity.inputsystem": "1.0.0",

I have the same issue, and the issue @ROBSCHERER123 mentioned with the mouse scroll wheel scrolling really really slow.

I added:
#if UNITY_WEBGL && !UNITY_EDITOR
zoom *= 100f;
#endif
to my code to work around the scroll issue for now, but I have not found a workaround for the Input Fields not working. I did notice that arrow and Home/End keys move the cursor around, but nothing seems to happen when typing letters or numbers (even from the numpad).

I am having this issue as well. Also adding that related to this is the mouse scroll wheel and using ScrollRect's in game. If you use the new input system and publish to WebGL, the scroll rect will scroll extremely slow. However it will work fine in the editor.