Well, obviously, if we’re talking “all available vocabulary” one can see how this might be so. It is said that English doesn't so much borrow words from other languages as follow them down dark alleys and beat them up to rifle their pockets for new words. English has borrowed words from languages it has encountered all over the world. It has also coined new words from Latin and Greek roots to describe things in the scientific realm. And English has the largest recorded vocabulary of any language in history. No other language even comes close.
But even the most fluent and erudite users of English don't know or use all the words available to them at any given time. For that matter, it matters what one is talking or writing about. A scientific or theological work would use a lot of words with Classical roots. A work on law or government would use a lot of words with Romance roots (esp. Latin and Norman French).
So, if we were going to test this hypothesis, the sort of work selected to analyze would matter. I decided to try; accordingly, I selected a non-specialized LJ post I had made recently. It was mere narrative, given in my native dialect. No effort had been made to use or avoid any particular words. I took the title and the first paragraph and did an etymological investigation of every word. (If you're wondering what text I selected, you can find it below, in the analysis.)
The first problem I encountered in my analysis was determining what counted as a "word." St. Patrick’s Day is a proper noun, but actually three different words. For that matter overalls is a single word, but made up of two others. I decided to count as a word all the source words.
And in counting words, a second problem arose: do we count numbers of entries only, or total word usages? Obviously, some words are repeated, and some appear in different forms ("class" vs. "classes"). I decided to do it both ways and see how it came out.
Third, what do we do with words that come from one language, but whose ultimate root is in another? When in doubt, I gave all the advantages to the non-English sources. Since my prejudice for Old English roots is the start of my challenge to the myth that non-English sources predominate in English, I must bend over backwards not to show partiality.
1. OE was the source of a majority of word entries. This was true even when I counted "spangle" as a Dutch loan word (I could have made the case for its source in English: it was borrowed in from Middle Dutch in the 16th Century, but MD spange is a direct cognate with OE spang; both meant "brooch, ornament.") I was surprised to find no Old Norse words in my sample. (I was also surprised at the absence of any Spanish or German sources, but less so.) Now, the native English majority was barely half, so it’s conceivable that another non-specialized sample might show OE below half, and a sample from a work on, say, science, theology, or law would give the impression that OE was the source of only a plurality of word entries. But from this one, non-specialized sample, the myth of the non-English origins of English has received a blow.
2. When considering word usages, (that is, accounting for repeated words) OE was a solid majority of all word usages. 60% and up would be considered a landslide if we were counting votes. So much for that myth.
And it didn't hurt me a bit. I mentioned in a Facebook post having learned “Now Thank We All Our God” in elementary school Music class. The school taught us all sorts of things belonging to the common culture, including religious songs. I learned to sing the full Malotte arrangement of “The Lord’s Prayer” in those classes. We also learned various patriotic songs (e.g., “The Star-spangled Banner”), and a mélange of ballads and pop songs, including “Here Comes Santa Claus” for Christmas and the now politically incorrect “Who Threw the Overalls in Mrs. Murphy’s Chowder?” for St. Patrick’s Day.
Total entries 75
Entries from OE 40 (53.33%)
Entries from Dutch 1 (1.33%)
Entries from French 16 (21.33%)
Entries from Latin 14 (18.67%)
Entries from Greek 3 (4%)
Entries from Irish 1 (1.33%)
Total Germanic entries 41 (54.67%)
Total Romance (French + Latin) entries 30 (40%)
Total Classical (Latin + Greek) entries 17 (22.67%)
Total other entries 1 (1.33%)
Total word usages 103
Word usages from OE 62 (60.19%)
Word usages from Dutch 1 (0.97%)
Word usages from French 16 (15.53%)
Word usages from Latin 20 (19.42%)
Word usages from Greek 3 (2.91%)
Word usages from Irish 1 (0.97%)
Total Germanic usages 63 (61.17%)
Total Romance (French + Latin) usages 36 (34.95%)
Total Classical (Latin + Greek) usages 23 (22.33%)
Total other usages 1 (0.97%)