I don't have the missal handy, but I believe we chanted the vidi last year at our '62 triduum.

You need to look in the OSH and the simple version of the same book in addition to the missal.

Ordo Hebdomadae Sanctae. It is the official book for Holy Week promumgated for the celebrations of 1956, so it also is OHS Instauratus, “restored.” There is another version if the rite is done without deacons and subdeacons.

A university library might have it if there is a Catholic one nearby or in the system, if you can get an ILL... Otherwise, you might have to email the ICRSS, the FSSP, or some other knowledgable folks. I can give you names if you need them.

The reason Vidi aquam is weird is because, unlike in the Novus Ordo, it has the versicles and collect. Doing the rite in silence is also bizarre.

I’ll refrain from going too far off topic, but this is one reason the Pian Holy Week is, in the whole, a disaster. I hope you find an answer...

I have dug out one of my copies of,
Officia Nova Hebdomadae Sanctae 1956, No. 851c
My copies have the Latin Rubrics and a translation in English on the opposite page.

"The Sacrament of Baptism
22.[...] All then go back in silence to the choir, and the renewal of Baptismal promises begins.
23. If however the Baptistry is separate from the church and it is preferable to bless the water in the baptistry itself, [...]
The cantors and people remain in their places and continue the Litany. If necessary, they repeat the invocations from Sancta Maria, ora pro nobis.
[...]
After the blessing all return to the church in silence, to begin the renewal of Baptismal promises.

25. [...] Then the Priest sprinkles the people with the holy water that was put aside, as said above, [...]

27. After the renewal [...] begin the second part of the Litany, [...]"

Well it does not suggest that something is sung during the sprinkling, and it should be noted that it is very clear about what is done... not that I can see anything wrong with singing or playing something.

Anyway my advice would be to use pre-1955, you know it make sense!

Madorganist is correct: since the blessing of water takes place in the vigil, i.e., before the Festive Gloria and the three-fold alleluia, Vidi aquam may not be appropriate, because of its inclusion of the double alleluia at the end. The alleluia is integral to this chant, since its text ends, "and they say alleluia, alleluia."

Clearly, the extraordinary rite does not provide for any music during the sprinkling. It would be worth it to reflect on why this is. It could be that the rite is complicated enough as it is that yet another piece of music is not foreseen, but there may be other reasons.

The rite does provide for a procession to the font, during which the tract Sicut cervus is sung. In places where there is no separate font, there is no need for this chant; so I have always sung it during the sprinkling, and actually we sing Palestrina Sicut cervus, which lasts about the same duration as the sprinkling. The rite does not provide for this, but neither does the rite forbid the singing of a motet in addition to what the liturgy provides.

Sorry I have no answer but I would say listen to @mahrt.
But I have a question too.
On Sunday I sang the vidi aquam start of mass and the priest was done before I could get to the verse part. Do I still sing Gloria and back to vidi or not?
He didn't mind, he said it was perfect but I don't think he was thinking of my singing at that point rather "where did all my servers go, I need this book here, and that candle there and the thurifer looks like he's not ready to come see me." As he was dealing with a whole team of servers and for them it was their first time serving mass.

To avoid confusion: For OF Asperges it looks from GR 1974 as though only the antiphon is required, the verse is ad libitum.