Learn Italian: Visto che manca la matricola, l'ammissione è impossibile. - Since the student ID number is missing, the admission is impossible.

Not in this exact structure. A present participle can replace a relative clause (for example, replacing che manca with mancante to say la matricola mancante for "the missing ID"). However, visto che ("since") introduces an adverbial clause of reason, not a relative clause modifying a noun, so it requires the conjugated verb manca.

Why use visto che for "since"? Could I use da quando?

Visto che literally means "seen that" and is used for "since" when providing a reason or cause (just like dato che, siccome, or poiché). You cannot use da quando here because it only translates to "since" when talking about the passage of time.

Why is the word order manca la matricola instead of la matricola manca?

While both are grammatically correct, Italian heavily favors placing the subject after the verb when dealing with verbs of existence, arrival, or absence. Because mancare describes a lack of something, putting the subject (la matricola) at the end sounds much more natural.

Is matricola the common word for any ID number?