He was born in Beverly, Massachusetts. In 1835, he joined the employ of the Columbia River Fishing and Trading Company of Nathaniel Wyeth. In December he arrived at Fort Hall in present-day Idaho and joined his first trapping expedition with experienced mountain men the following December. Much of what is known about this portion of his life comes from the journals of his travelling companion Osborne Russell. In January 1838 he camped at Jackson Hole with Jim Bridger and spent the next year acquiring a large number of beaver pelts in the Yellowstone area. In August 1839, he became separated from his party, which waited for him for several days until threat of an attack from the Blackfoot forced his party to return to Fort Hall. The following month he returned to Fort Hall by himself unharmed.
On August 22, 1842, while in the Snake River valley, he and Osborne Russell joined a wagon train led by the missionary Dr Elijah White headed the Willamette Valley. While serving as a guide for the wagon train, he met Hannah Able, a young widow from Indiana with a baby daughter travelling with the W. T. Perry wagon. On arriving at Willamette Falls at present-day Oregon City, the two were married on October 20.
Elbridge and Hannah set up a homestead Clatsop Plains near Astoria at the mouth of the Columbia River. In 1852, they left the Clatsop Plains to settle near Tillamook Bay south along the coast. They were the first white family to settle in the bay, establishing a homestead along the Trask River, which is named for him.
In 1960 Trask was popularized in the historical novel Trask by Don Berry. The novel, as well as its two sequels, are collecively known as the "Trask novels."