I usually log in to a Stack Exchange site using Facebook as my OpenID provider. However, providers such as Gmail, Yahoo and Facebook are blocked. Is there any way for me to log in to my existing Stack Exchange account even though my account provider is blocked?
Yes, you could use Stack Exchange's very own OpenID provider (or any other OpenID provider your company doesn't block), and associate that new ID to your account by following: How do I change my OpenID provider(s)?.
If you are currently using one of the "most trusted" providers (Google, Yahoo and Facebook at this time), and use the same email address while registering, account association well happen automatically, so you don't even have to worry about that step.