The GIRM is referring to a song as a chant or sung text.
However, in the ordinary form there is the absence of the offertory antiphon in the missal yet it still exists in the graduale.
I suppose in the nearly anything goes ordinary form it is difficult to say whether there is truly any legislation regarding a purely
instrumental offertory, since the Mass itself could be sung or not sung.
The preferred approach in your case, would be to chant the offertory antiphon (RG, SEP, Simplex or Flowing Waters) then allow an
instrumental (organ or otherwise) to accompany the rest of the liturgical action.

The new GIRM is included below. Taken all together, ICEL and the Bishops collectively want to emphasize the text, chant as the preferred way of emphasizing that, and congregational participation. Since even the word "song" has now been eliminated, it would seem that an instrumental offertory is counter to the new instructions.

As the Latin essentially has a blank slate for Episcopal Conferences to fill (and therefore there is no appeal to the Latin text from the English text), this is what they want. However, ultimately, it comes down to what your Bishop allows (most likely, not through active legislation) and even more your pastor. But here is what the bishops say...

37. Finally, among other formulas:
a) Some constitute an independent rite or act, such as the Gloria in excelsis (Glory to God in the highest), the Responsorial Psalm, the Alleluia and Verse before the Gospel, the Sanctus (Holy, Holy, Holy), the Memorial Acclamation, and the chant after Communion;
b) Others, on the other hand, accompany some other rite, such as the chants at the Entrance, at the Offertory, at the fraction (Agnus Dei, Lamb of God) and at Communion.

These are the norms for the Entrance, which the GIRM says are the same as for the Offertory:
48. This chant is sung alternately by the choir and the people or similarly by a cantor and the people, or entirely by the people, or by the choir alone. In the Dioceses of the United States of America, there are four options for the Entrance Chant: (1) the antiphon from the Missal or the antiphon with its Psalm from the Graduale Romanum, as set to music there or in another setting; (2) the antiphon and Psalm of the Graduale Simplex for the liturgical time; (3) a chant from another collection of Psalms and antiphons, approved by the Conference of Bishops or the Diocesan Bishop, including Psalms arranged in responsorial or metrical forms; (4) another liturgical chant that is suited to the sacred action, the day, or the time of year, similarly approved by the Conference of Bishops or the Diocesan Bishop.

Mr. Thompson,

Respectfully sir, I have been to the Novus Ordo Mass for all 29 1/2 years of my life. I can say now, without hesitation, that I have NO DOUBT why people desire the TLM - it's in what "we've done AND in what we've failed to do" all of these years. I mean, why do we have to have an array of folks desiring to understand what these things mean? Shouldn't we be allowed to expect that the text will clarify and handle almost every situation? The ambiguity left, usually to coddle folks who will knowingly be anti-authority anyway, is disheartning and silly. The same reason probably covers why we have gotten rid of Holy Days of Obligation and why we have gotten rid of a mandatory Friday abstinance from meat. "People won't do it, anyway." So let's make our documents as loose as possible, knowing that people won't give one flip about 'em anyway.

Some of these folks who write these things should come teach, like me, in an elementary classroom. With rules this ambiguous and utterly ungrounded, their classrooms would resemble a zoo. I can see the look on the kids' faces, "What's he saying? I don't get it..."

JIF: since the musical rubrics of the EF are so clear and do not contain any "disheartening" ambiguity "to coddle folks who will knowingly be anti-authority anyway," perhaps you can apprise us of whether, and under what conditions, instrumental music at the Offertory was permitted. After all, we wouldn't want a hermeneutic of rupture between what was allowed in 1958 and what is allowed today, would we?

It's a similar situation to a book on grammar: should it be DEscriptive or PROscriptive? In other words, should it tell you how you actually DO speak, or how you SHOULD speak?

So many Church documents tend to be descriptive, possibly in the hope of not excluding existing practices. Most of us are tired of this, and want proscriptive documents.

You could do a brief vocal offertory, and then organ music. But it seems to me that "organ offertory" is a logical impossibility, given that organs cannot enunciate text. (Yes there are "organ masses"; they're a liturgical abuse.)

Yes, I know there's a long tradition of alternatim settings. Maybe it's the unreconstructed Lutheran talking, but the Word is the word. If the text is very familiar and there's a clear tune (e.g., alternations in the Kyrie), it's not so bad, because the congregation will play back the words in their heads. And maybe it doesn't go all the way to illicit...but it's still not Best Practices.

LOL! My Lutheran liturgical experience begins and ends with the 1940 hymnal, but I thought the Lutherans had been buying all the bad liturgical ideas pioneered by Catholics post V2.

I was SO Misery Synod, but that was pre-1973...as my sainted grandmother said of ELCA, "Why, if they believe that, they aren't even Chris-tee-an!" My last experience of Lutheran liturgy was ELCA though...after trying to get the congregation to sing a very boring African hymn, the pastor said, "We'll have to work on that one." Hint: when Lutherans don't sing, there's something seriously wrong.

perhaps you can apprise us of whether, and under what conditions, instrumental music at the Offertory was permitted.

Actually, it was quite common for an organist to play during the EF Offertory, and my memory of that goes back to the early 1950's. For a Missa Cantata, the schola would sing the Offertory verse first (of course) and if the choir sang a voluntary, the organ usually did not play afterwards. For a LOW Mass, however, instrumental (organ) music was usually played through the saying of the prayers. Matter of fact, there are hundreds (or maybe thousands) of organ pieces entitled "Offertorium." There's a reason for that.

The only rule was that the music should not make the priest stand around and wait for the music to stop.

I completely agree with Benedictgal.

Maybe the world IS coming to an end ...