At least as of 2018, it does not seem as if the original Spanish text has been published, only a criticized Italian version and an even more criticized English version.

Here's an Associated Press article from 2018, when the controversy first erupted after the publication of the English version, which was a translation not of the original but of the Italian (the Italian version was original published in 2005, I believe): "Key findings in analysis of memoir of a Jew raised Catholic".

Recently, the case has made headlines again after a U.S. historian, David Kertzer, found discrepancies between the Spanish text of Mortara’s memoirs held in the archives of his religious order, and an Italian translation published in 2005 by Italian journalist Vittorio Messori.

Another article, published in the Atlantic and written by Kertzer himself, mentions the specific archive the manuscript lies in: "Rome’s Canons Regular archive".

In 1888, three decades after he was taken from his family, Edgardo wrote an account of his story in Spanish, referring to himself in the third person. By then, having been raised in a seminary, he had become a Catholic priest; years later, members of his order prepared a typescript of his account. It was the copy of this spiral-bound memoir in Rome’s Canons Regular archive that Messori says he found and translated into Italian for publication; the English version that appeared last fall relies on Messori’s translation, rather than the original Spanish. I personally compared the Italian and English versions of the Edgardo Mortara memoir published by Messori with the original, which I too located at the Canons Regular archive and digitized.

So it sounds like it is only available in Spanish in manuscript form. I assume if the text had been available published in Spanish, he would not need to travel to the archives, or else it would have been at least mentioned here (something along the lines of "Vittorio Messori's Italian translations which was the basis for the English does/doesn't match the published Spanish version"). This being the Anno Domini 2018, it does sound like he not just looked at it but created a personal digital copy.

What you may do is simply write a polite email to Professor Kertzer, explain your interest, and ask him if he might be willing to share his digitized version of the original Spanish with you or upload it to Archive.org or some place similar publicly accessible depository. He may not, but short of traveling to Rome and visiting the archive yourself (and as far as I know, Vatican archives generally demand academic or journalistic credentials), that may be your best bet to read it in its original.

According to Vittorio Messori, who published Mortara's memoirs in 2005 - Io, Il bambino ebreo rapito da Pio IX - the published notes reproduce a version from Spanish made by Andrea Vannicelli, which the author (Messori) has deemed accurate. This is the source for the subsequent English version by M.J. Miller published in 2017. I could not find a published version of the Spanish one, even if it would appear that the document is available for examination, with the necessary credentials.

It would also be reasonable to expect a precise indication of the original's location in the Italian published translation. This doesn't exactly appear to be the case... Anyways, Mortara's personal papers appear to have been kept at Rome’s Canons Regular archives. There, they were located by Messori first - who proceeded to the publication (a work, at least in part, dedicated to rebuke the polemics surrounding the canonization of Pius IX in the year 2000) - and by David I. Kertzer then, who compared the original with the published version (at least in part due to his ongoing polemics with Messori) finding significant alterations. Here, if you like, is Kertzer's take on the matter.

It would appear that the manuscript - regardless of certain translation liberties - has undergone a degree of editorialization, to make it more palatable for modern audiences. Which, if I am allowed to express an opinion, is rather poor form for what purports to be the first (accurate) published version of a newly discovered document. For instance, according to a report from the Associated Press, some antisemitic remarks included in the original have been removed.

One can understand why, as back in the day - the manuscript was written in 1888 - one would not be surprised to see the idea that the Jewish religion was "absurd" and "ridiculous" as explicit part of a Catholic education. This would, nonetheless, undermine the purpose of the memoir that is to justify the measures taken by the Papal authorities and especially by Pius IX against the Mortara family and - ostensibly - for the sake of the child Edgardo.

I have a feeling that the modest benefits coming from this omission are largely squandered by the unfortunate foreword by R. Schoeman who informs us that:

Promoting the welfare of its citizens has always been seen as a legitimate concern of the state, perhaps the primary one. Throughout the United States and Europe today, the state is considered to have the right even to remove the chid from his parents to protect the child's physical and emotional well being [...]

Only temporal - not eternal - welfare is usually considered.

But what if the teaching of the Catholic Church is true? What if, once created, the human person lives for all eternity, and the nature of that eternity - whether perfect bliss or unending misery - is dependent on the sacraments and on the person's moral formation? [...]

It is true, as Schoeman argues, that the controversy took place at a time when the state's laws and institutions - and in part society alltogether - were moving towards a separation of secular and temporal principles. A manifestation - if you will - of that "misnamed Enlightenment" leading to the "rejection of Christianity" and to the horrors of "homicidal, anti-Church French revolution" and "atheistic communist revolutions that have plagued the world since the start of the twentieth century".

The fact is that all that - I dare say - has close to nothing to do with the particular history of Edgardo Mortara, who was - after all - a small child taken from his family. I have no particular fondness for anti-clerical or anti-religious polemics: it's drab, dire, and often little more than an effort to deman and belittle people who have a genuine desire to commit to positive beliefs and to the idea of a shared community. But I am sincerely perplexed by a foreword that seems to argue that forced conversion of children is the moral choice, if it saves them from eternal damnation. Nor this appears to be the official position of the Church. And, in fact, I dare say that it was not - striclty speaking - the official position of the Church in the mid XIX Century either. That's where the line between anti-semitism, anti-judaism, and the legal prerogatives of a confessional state begins to blur. Mortara had seven siblings: it was specifically his prerogative as a "Catholic" that made him object of the special attention of the state for his welfare; it was specifically the violation committed by the Mortara family in hiring a Catholic that left them no legal recourse on the matter (which they might have had, had the child been Christened surreptitiously). The context of this tale is one of specific discrimination of Jews within the Papal States; a discrimination that derived in part from concerns over the spiritual welfare of the Pope's subjects, but that can't really be framed exclusively that way.

Edgardo Mortara - for those who don't know - was a child, born in Bologna in 1851, then part of the Papal Legations. In 1852 he had fallen severely ill. A young housemaid - a Catholic, Anna Morisi, then aged 16 - had believed the child to be about passing away. As was (legal) custom, she had then recited the ritual formula which allowed the Christening of people in danger of imminent death.

Five years later - the child had survived - another housekeep of the family, who had heard it from Morisi, appears to have reported this fact to the authorities. The inquisitor of Bologna - Pier Gaetano Feletti - after confirming this fact with Morisi, had sent the police to take the child from the Mortara family.

This, being the Papal States a confessional state, was both a religious and a legal matter. First, the Mortara family had violated the law by hiring a Catholic as housemaid. Then, they had unwittingly violated the law by raising a Christened child as a Jew. They could not be held at fault for that; but they also could not be allowed to persevere. Edgardo was then taken to Rome, and transferred to a religious institution, where he was soon visited by the Pope himself. The Pope - on this point everyone seems to agree - was genuinely concerned with the spiritual welfare of the child, and committed to ensure him the Catholic education necessary for his salvation. Unfortunately, this meant enforcing the separation of the Catholic child from his fomer Jewish family.

The case was not unique, to the point that this was a somewhat standard proceeding as maintained by the Papal authorities. But it took place during a period of social and political unrest which had seen the Pope - originally regarded as a "progressive" and liberal figure (once again, Pius IX was genuinely concerned with the well being of all his subjects) - taking an increasingly more resolute stance against liberalism, secularism and, by extent, against the process of Italian unification. Consequently, in their efforts to regain custody of the child, the Mortara family received the assistance and sympathy of large portions of the liberal establishment both in Italy and in the outer world. The case provoked a certain echo in France, England, and in the US; but despite this mounting pressure, the Pope would not surrender the boy.

Indeed the Catholic press fought back against the accusations of obscurantism and abuse, claiming that the young child had soon taken to refer to Pius IX as his real father, and showcased a sudden interest in the Catholic religion that showcased the miracolous effects of the Christening he had received.

There is no doubt that the education Mortara received after his removal from his family influenced his views on the matter. Mortara did - unquestionalbly - spoke fondly of Pius IX, regarding the Pope as more than a spiritual father, and openly acknowledgling that his intervention had saved him. He did commit to the Catholic Church, becoming a priest and an active figure in spreading the Catholic faith. His contacts with his family instead remained sporadic. After turning eighteen he refused a first offer to rejoin them, and would later entertain a correspondence with them largely to encourage them to convert.

There is also little doubt that - ultimately - his voice on the matter should be listened. Mortara had chosen the name Pio Maria for himself in homage to his father Pius IX, and his memoirs - particulars aside - was written to defend and justify the actions of the Pope and the position of the Catholic Church.

At the same time, his family's recollection reveals a slightly different picture: at least in their early meetings, under surveillance, the boy had appeared in distress, and had begged them not to leave (which, to be fair, would appear slightly more consistent for a seven years old who had just been taken from his family).

I doubt the original document would give us the truth of the matter, save for the fact that people will often commit to positions that are cruel and harmful to others, even out of what they perceive as the true and superior justice.