I can't say that I know the origin of this unusual name, but it would be fun and perhaps instructive to try to make an educated guess. All of my Sheppeck ancestors came from Dorset, England, and particularly the area around Bridport, so that is a place to start.
The name has many variations, mostly spelling differences involving the vowels, and the "ck" at the end is sometimes a "ch." The oddest variation I've found is Shakeup, which was probably a mis-hearing of the name by an official. But the variation that I think points to an origin is Shapwick, which in England would be pronounced something like Shappick.
Shapwick is a Saxon placename meaning sheep farm. Sheep are certainly plentiful in all of Saxony, which is the southern and western part of England, so this is potentially a common placename. I have only found two villages ever called Shapwick, however; the more well-known one in north-central Somerset, and another in eastern Dorset. A family from Shapwick could have called itself "de Shapwick," and this could have devolved to Sheppeck.
Some evidence would come in handy, and indeed there are historical references to de Shapwicks from both of these villages. But the name seems not to have stuck as a surname. There is little evidence of any clan of Shapwicks or Sheppecks around either of these villages, or anywhere else in England with one exception. There are plenty of Shapwicks, Sheppecks, etc. in western Dorset going back to the mid 1600's, in Allington, Bridport, Beaminster, and the surrounding towns.
Could this group have moved here from one of the Shapwick villages? It's possible, but I don't think likely. There are no direct routes connecting either Shapwick with this part of Dorset. Could a nearby manor or estate have been called Shapwick? I began digging deeper into old placenames, and sure enough, I found a Shapwick estate quite nearby, just over the line in Devon.
Long ago English counties were divided into administrative units called hundreds, and each hundred was divided into ten tithings. Shapwick Devon was once a tithing of Axminster hundred, and a part of the manor of Axminster. Now Shapwick is just a part of the parish of Uplyme. It lies west and a bit south of Uplyme village, about 4 miles south of Axminster town and a couple of miles northwest of Lyme Regis, in Dorset. The estate ranged from Shapwick Hill southward to the Lyme Regis - Exeter highway. In his history of Devon, Hoskins says that Shapwick is mentioned as early as 1167.
Finding Shapwick on old maps is fun enough, but there's even a book! It turns out that a Cistercian monastery in Axminster called Newenham Abbey ended up owning Shapwick. The details are laid out in a book called The History of Newenham Abbey, written in 1843 by James Davidson, and this book is available and viewable online thanks to Google's book-digitizing project.
In 1245 the manor of Axminster was given to Beaulieu Abbey in Hampshire for the purpose of establishing a new abbey. It was built about a mile southwest of Axminster town, above the Axe River, and named Newham. Some of Shapwick was part of this gift, and within a few years the Abbey had acquired most of the remaining portions of Shapwick.
The last part, donated in 1333, was a part once held by a family named "de Shapwick." It wasn't clear even then who this family was, so the Abbey asked for and received a brief pedigree, as follows:
The last Shapwick to hold the land was John, in 1317. His son, also John, became chaplain and prior of St John's Hospital in Bridport, Dorset, in 1357. So there's the Bridport connection, and perhaps the ancestor of all us Sheppecks.
Devon County History Page
The History of Newenham Abbey