Can I configure Oracle such that a Windows client can connect to a Linux Oracle server without a password?
Kerberos is way too complicated for this task, I would not bother with it.

ODAC supports using the a Secure External Password Store:

Using Secure External Password Store

You can point to the wallet location in your configuration in the settings section .

An example:

https://dbaportal.eu/2015/11/26/odac-12c-release-4-odp-net-managed-driver-with-oracle-wallet-gotcha/