This is a browser issue, just open the authorization page in a new tab / windows.

NO, it's not.
This is an application issue, not a browser issue. The browser only does what it's told by the application. In this case, the application developer (you) are telling the browser to refresh too quickly.

The browser is following the instructions sent by the application, which is sending invalid instructions.

I use the regular redirecting function in vBulletin! exec_header_redirect()

I've tried the javascript one, didn't work too!

It's something like the browser doesn't like to open another website inside iframe.

Why do the same instructions that I send to the browser do the job well when we open the authorization page in a new tab?

You're the designer, you need to answer your own questions.
The facts:
You're redirecting users via whatever method.
That redirection is causing issues. The BROWSER isn't causing the issues, the APPLICATION is. HOW you redirect your users is causing problems, fix it, or remove the redirect.

What's the big deal? This isn't preventing the application from being usable or not.
You only have to use this page once, and the solution is to open it in a new window.

No, that's not the solution, that is a workaround of the problem created by poor coding, nothing more.

What's the big deal? If the developer can't find and fix this simple problem, what assurances do the rest of the users here have that they'll fix larger ones?

wow... After reading your comment.

Dude your rude.

I'm glad we didn't proceed after you accepted the work and then ignored my communications on the project.

I wouldn't tolerate that kind of speak. The developer here is offering a free service and is communicating with his peers throughout the development. Maybe start building your own mod instead of picking apart others work.

That was very uncool.

I was going to say something of the same. Be glad he made the mod and released it for free. It's working great on my forum and appears to be coded very well.