While the vast majority of pages citing the Trump-Drumpf link seem to be citing the book The Trumps: Three Generations That Built an Empire by Gwenda Blair, she seems to be saying different things at different times regarding this issue.
This page describes the family as having changed their last name while they were residing in Kallstadt during the Thirty Years' War in Germany. (Transcribed from a screenshot of the preview of the book on Amazon.com.)
In 1608 Hanns Drumpf, an itinerant lawyer, showed up in Kallstadt, then a muddy settlement of fewer than six hundred people. Most were involved in winegrowing and did not welcome newcomers into the vineyards. But soon after Drumpf’s arrival, the Thirty Years’ War engulfed Germany and ushered in one of the worst periods in Kallstadt’s bloody history. The village burned to the ground at least five times—the period was so chaotic that nobody is sure of the exact number—and at one point only ten families remained in residence. By war’s end in 1648, about 40 percent of its inhabitants had died, and economic activity came to a standstill.4 It was a scene of near total devastation; it was also an opportunity, however unbidden, for Drumpf’s family to gain entry into Kallstadt lift. By the end of the century a winegrower named John Philip Trump—the family changed the spelling of its name in the course of the war—was a taxpayer in good standing. Although their holdings were never more than modest, the Trumps had become part of the village’s social elite.
However, the author later reversed her position during an interview with Deutsche Welle, which stated that:
Gwenda Blair: His grandfather Friedrich Drumpf came to the United States in 1885 which was the height of German immigration to the United States when he was 16. His family was from Kallstadt, winegrowers.
Therefore, barring any additional evidence (such as the immigration papers of Friedrich Trump/Drumpf), the originator of the claim has contradicted herself, and is unlikely to be considered a reliable source.
The recent immigration papers that appeared on Wikipedia are also suspect - the original image upload did not cite a source for the image, and scans not produced by actual archival organisations are likely to be at risk of tampering. Without evidence that the papers came from a reliable source, it is unlikely that these papers would settle the issue.
However, another recent addition to Wikipedia has an image of a passport application form signed "Fred Trumpf" which is claimed to come from the paywalled site ancestry.com. If this document is shown to be genuine, it could be possible evidence that settles the issue.