Is it permissible to invoke blessings upon the Prophet, may Allah bless him and grant him peace, in the form of "Allahumma salli 'alayh" (O Allah, bless him) or "Allahumma salli 'alayhi wa sallim" (O Allah, bless him and grant him peace), and to repeat it?

Praying upon the Prophet, may Allah bless him and grant him peace, is an act of worship with immense reward, and it can be performed in any form. The best form is the Abrahamic formula. Therefore, it is permissible to pray upon the Prophet using the formula mentioned in the question.