Although "aspirandato" exists, it is much more rare than "aspirantato", meaning the same thing. Common in religious terminology but rare otside it.
Here are examples for both, to show that they are the same (Treccani has a definition too):
1.
TABOÃO DA SERRA - CASA DI ASPIRANDATO
www.sacrocostato.org/dove...aspirandato/ruteng_aspirandato.... Suster-Suster Misionaris Lambung Kudus Yesus Jln. Pelita, 15. Ruteng, Manggarai

2.
Salesian Termbase - Termbase - aspirantate
www.sdb.org/pmwiki/pmwiki.php?...Aspirantatate
26 Nov 2011 – In actual fact, especially in regions where vocations are plentiful, the aspirantate has made a comeback! The term 'aspirant' should not be ...
Regional usage:
Description:
The term does not appear in the Constitutions (but in the Regulations instead) or the Ratio. DB in his own time made reference to Apostolic Schools (Meeting of Superior Chapter, June 5, 1884), a model which he had heard of in France and which was a kind of junior seminary to prepare very young candidates for novitiate or seminary entrance.The fact that the term does not appear in the Ratio might suggest it is on the way out. In actual fact, especially in regions where vocations are plentiful, the aspirantate has made a comeback! The term 'aspirant' should not be confused with 'seminarian', A seminarian could be an aspirant but an aspirant may not be a seminarian. The term 'aspirantate' will normally be understood in religious circles (meaning Religious Congregations) but is rare outside that. It is obviously a derivation from 'aspirant'. In a joint document released in 2011 by the Councillor for formation and the Councillor for Youth Ministry (on the experience of the aspirantate), the term 'aspirant' has now a preferred target group, viz., young men interested in Salesian consecrated life who have already completed some post-secondary studies. This does not rule out other candidates but is rather a statement of preference.

Context (examples of use):
The term aspirantate will normally be understood in religious circles (meaning Religious Congregations) but is rare outside that. It is obviously a derivation from 'aspirant'.