I suppose it wouldn't technically be the Hundred Years' War if it weren't between England and France, but there's always the possibility of creating a year that lasts as long during about the same time period between England and the HRE. There would probably need to be an occurrence where the English have nobles that end up with holdings within the empire that threatens the Holy Roman Empire. The easiest place for something like this to happen would certainly be in the Low Countries. The English traded extensively there, especially with Flanders, so seeing Edward III marry Philippa of Hainaut makes a good amount of sense.
Anyway, one thing that could happen to start tensions could be the death of who would have become William II of Hainaut in OTL before his father, the reigning Count of Hainaut does. Assuming either no other sons are born, grow up past childhood, or die somehow before the old count, the oldest daughters of William I of Hainaut are Margaret and Philippa, who respectively married Louis VI, the Holy Roman Emperor (and a Bavarian Wittelsbach) and Edward III of England. In OTL, their brother William II died in 1345, so the county cleanly passed to Margaret, but only after Philippa and their other sisters gave up their own claims. Since some of the possessions of William I included Holland and Zeeland, which had some provisions that did not allow women to succeed to rule those counties (not a problem in Hainaut), this could lead to complications if Philippa, the only other sister married to a powerful monarch, refused to press her claims, and her husband Edward, at this point already looking for a reason to claim the French throne, is instead compelled or convinced to fight for his wife's claims instead.
It would be messy. Louis VI was an enemy of the House of Luxembourg, which produced the previous emperor (and in OTL, the succeeding one). If he goes to war on his wife's behalf in what would start off as a war for the succession of 3 important counties within the Empire that Louis can't afford to lose, Luxembourg would surely support Edward. The would be Black Prince could find himself engaged at a young age to foreign noblewoman or princess in the name of a formal alliance.
Phillip VI of France, the uncle of the Hainaut sisters, could sit back and wait for his own moment to claim part of the spoils or decide who he wants to support. Any chance to see the HRE lose imperial territory in the Low Countries is always good for France, but at the same time, Edward had been railing about his own claim to France, so Philip could just watch his two rivals beat each other up in the north, while demanding quite a bit for his neutrality from both sides.
The current emperor can't just give up his wife's claim. He absolutely needs to look strong here if Edward and Philippa continue to claim Hainaut, Holland, and Zeeland, all while the House of Luxembourg would fully throw their support behind England to further pull him into their own camp against Louis and the House of Wittelsbach. The rest of the empire may have to pick sides here between the two rival camps.
