Can institutions agree to submit proof of application at the party?

You can't. According to Tuhua Education Network, in order to ensure the consistency of open recruitment policies in various provinces and cities, the recruitment work of institutions in Anhui Province requires that in the qualification examination, the official staff of government agencies and institutions must also provide the certificate that the unit and the competent department agree to apply for the exam according to the cadre and personnel management authority, and the institution shall not delay submitting the certificate that agrees to apply for the exam.