At the time of European contact, what is know southern New Jersey and southeastern Pennsylvania was entirely populated with Unami-speaking Lenape communities. At the same time, the Shawnee occupied the middle Ohio Valley and neighboring tributaries, but between the mid-1600s and the early-1700s the Shawnee diaspora scattered the Shawnee throughout the eastern part of the continent. By the 1690s there were at least two Shawnee communities on or along the Delaware River. In 1677, the Haudenosaunee (Iroquois Confederacy, of which the Seneca were one of the five members) defeated the Susquahannock (after whom the Susquahanna River is named) and claimed the bulk of Pennsylvania as their own territory. The Seneca, as the Keepers of the Western Door, were frequently the ones in charge of such military campaigns and the Seneca Trail which ran down through Pennsylvania as part of the Great Path was well known to the colonists. As the European colonists became more numerous in the area, the Lenape and Shawnee were pushed west. The Haudenosaunee granted them land in their newly conquered territory as tributary populations (the Haudenosaunee didn't have the numbers to populate all the land they were conquering so they instead gave it to displaced populations in exchange for their fealty).

The Cherokee homeland is in the Smoky Mountains, but at their peak they possessed a sizable territory - which you'll notice does not include any part of New Jersey or Pennsylvania. Sequoyah was a famous Cherokee smith who devised the Cherokee writing system. He never lived in Pennsylvania or New Jersey. Though he traveled extensively, I can't find any reference to him having even visited Pennsylvania or New Jersey. Of course, the school in question spells the name like the tree (which was named after the man), so maybe the school is really honoring some trees in California.

The Lenape (or Delaware) tribe absolutely was in that area. Seneca are part of the Iroquois Confederacy, so territorial claims (and actual territory) vary widely. The Confederacy is from upstate New York, but they certainly moved into Pennsylvania and claimed dominion over the Lenape at one point. When I think Shawnee, I think Tecumsah, so Northwest territory (Ohio, Indiana, etc.), but they apparently had a fairly wide territorial range. I'm not finding a Sequoia tribe. Someone else can chime in and correct me, but Sequoia is a Cherokee person. The Cherokee had a wide range as well, but I don't think their territory ever reached that far north. So I'm going to go with the latter (I'll skip the editorializing, but lumping of disparate peoples sounds about right).

My source right now is Wikipedia, although I've used other sources in the past, so I used it mainly to refresh my recollection. The Ambiguous Iroquois Empire would certainly cover the Iroquois and Lenape and gives you a good overview of Iroquois territory both real and theoretical. I don't recall much about the rest.