I usually log in to a Stack Exchange site using Facebook as my account provider. However, I'm currently in an environment where Google and Facebook are blocked, or I'm experiencing a bug with the mobile apps where Google login isn't working. Is there any way for me to log in to my existing Stack Exchange account even though I can't use my account provider?
Yes, you can use standard email and password login, and associate that to your account by following: How do I change my login provider(s)?.
If you use the same email address while registering, account association well happen automatically, so you don't even have to worry about that step.