Can a government company block the resignation process on grounds of vigilance case or internal proceedings?

Hello, I am working in a govt organization and things are not going well in my office. There might be some vigilance case pending against me. Can I resign and join a private company in this situation? Or do I need a proper relieving letter for joining a company?

Also, can a company block the resignation process on grounds of vigilance case or internal proceedings?

No one can be forced to continue working for a company he / she does not want to work for.
So no one can 'block' your resignation.

However, if there is a vigilance or internal enquiry going on, it will continue and you will be required to cooperate and provide the details as well as appear before the officer for hearing or clarification that may be needed. If you do not do so, it is possible that it may become a police matter, in which case, your new employer is likely to remove you for the job

If there is a Vigilance Case pending or is contemplated then the resignation is not accepted in Govt/PSU service. These are the HR Policies of the respective organisation and this is followed. So there is every possibility that your resignation from service may be rejected. Whether a Private firm will want relieving order of the previous employer for joining it depends on its policy as there are both types of organisation.