Abram vowed to take nothing from the king of Sodom, not even a thread or what?

Answer: A sandal strap

Abram said he would not accept "even a thread, or a strap of a sandal" (KJV: "shoelatchet"), lest the king say he made Abram rich.

“But Abram replied to the king of Sodom, “I have raised my hand to the LORD God Most High, Creator of heaven and earth, that I will not accept even a thread, or a strap of a sandal, or anything that belongs to you, lest you should say, ‘I have made Abram rich.’”— Genesis 14:22-23 (BSB)