It's entirely untrue. Copies of sections of the Bible, Bible storybooks, the weekly and feast day Gospel and epistle readings for Mass, and even entire Bibles that belonged to medieval lay owners survive today. The last part was rather rare before the invention of movable type, since the Bible is a long book and copied by hand that would be enormously expensive, but rare does not mean nonexistent and it certainly does not mean meriting the death penalty.

Where I suspect your source is getting confused is the so-called Oxford Constitutions of 1409, the pet censorship project of Archbishop of Canterbury Thomas Arundel. Virulently opposed to a bogeyman-like heretical movement the English derided as "Lollards" (followers of John Wyclif's ideas), Arundel presided over a decision to deny approval to any "modern" English translation of Scripture, at all. ("Modern" being "recent enough English that it could be understood by a contemporary speaker/reader). Article 7 of the Constitutions holds:

"Item, It is a dangerous thing, as witnesseth blessed St. Jerome, to translate the text of the holy Scripture out of the tongue into another; for in the translation the same sense is not always easily kept, as the same St. Jerome confesseth, that although he were inspired, yet oftentimes in this he erred: we therefore decree and ordain, that no man, hereafter, by his own authority translate any text of the Scripture into English or any other tongue ['aliquem textum sacrae scripturae auctoritate sua in linguam Anglicanam, vel aliam transferat'], by way of a book, libel, or treatise ['per viam libri, libelli, aut tractatus']; and that no man read any such book, libel or treatise, now lately set forth in the time of John Wickliff, or since, or hereafter to be set forth, in part or in whole, privily or apertly, upon pain of greater excommunication, until the said translation be allowed by the ordinary of the place, or, if the case so require, by the council provincial. He that shall do contrary to this, shall likewise be punished as a favourer of error and heresy." (version preserved in John Foxe's Acts and Monuments/Book of Martyrs)

Lay possession of vernacular (English) Scripture in 15th century England, then, could draw down accusations of heresy and even the death penalty. On the other hand, English Bibles from the time managed to survive. And more importantly, this particular strain of censorship was so severe only in England.

In Germany and the Low Countries in particular, the printing press churned out edition after edition of vernacular Bibles--many of which we know ended up in lay hands. There are some neat studies of how lay Dutch owners marked up their Bibles that show people took them to Mass and sermons with them to follow along and take notes for learning! In Germany, meanwhile, it's true that the Archbishop of Mainz in 1481 did declare that from then on, new translations of the Bible into German needed his explicit approval. This decision did reduce--reduce, though, not eliminate--the frequency of new vernacular Bible editions. However, none of the new ones were published by printers who bothered to seek approval. The only known case of actually getting approval comes from clear across the Empire, where the faculty of the University of Cologne apparently approved a German translation some years earlier.

Overall, the claim that lay people were banned from owning Scripture is not groundless for the fifteenth century--but it's still very much wrong.