Great question!
First we need to clarify a few terms. Tectonic plates are pieces of Earth's crust and uppermost mantle. They make up the entire surface of Earth. That means every place on Earth's surface is on a tectonic plate. What I think you are asking about are tectonic plate boundaries. Plate boundaries are where two plates meet, and these boundaries are where the most visible and spectacular consequences of plate tectonics are observed. When plates meet, they are either colliding together, pulling apart, or grinding past each other. That plate motion results in earthquakes, volcanoes, mountains, and giant rift valleys.

On the eastern coast of the US, there are not any current plate boundaries like there are here on the west coast. We refer to the east coast as a passive margin, because it was once where two plates ripped apart but no longer has active tectonic activity.

Great question! Indeed the east coast (of North America, I presume) sits on a tectonic plate, the same plate that eastern California and New York and everything in between sits on, the North American plate (which extends eastward all the way to the mid-Atlantic ridge).

Practically every place on our planet (under sea or on land) sits on one plate, or micro-plate, or another. The only exceptions are boundaries between plates, such as the San Andreas fault, which divides the North American Plate from the Pacific Plate, and mid-ocean ridges. Some landmasses, such as Iceland, straddle two plates. Indeed North America (geographically speaking) is composed of two plates (geologically speaking).

Have fun with Plate Tectonics!

Most of the United States, Canada, Mexico, and Greenland all lie on the North American Plate. Even part of Iceland and Russia lie on it! Because the plate includes a lot of seafloor in the Atlantic Ocean the east coast is pretty far away from any other plates.

Tectonic plates make up the surface of the whole earth. The east coast is located on the North American Plate, as are most of the rest of the United States, Mexico, Canada, Greenland, and part of Russia.

I started googling 'tectonic plates' and 'tectonic plates map' was one of the choices. Check it out - it's pretty cool. Tectonic plates are all along the west coast of North and South America. Also in the middle of the Atlantic Ocean - here's a link with a map that looks like other maps of tectonic plates in the google images:
tectonic plates.

The eastern boundary of the North American plate is the Mid-Atlantic Ridge, which is situated roughly halfway between North America and Europe (this is also where the European plate begins). This means that while, yes, there is an eastern edge of the North American plate, it is thousands of kilometers from the coast, instead of being very near the coast (northern California, Oregon, Washington, British Columbia, southern Alaska), on dry land (central and southern California, northernmost Baja California), or even separating a peninsula from the mainland (Gulf of Cortez) as it is on the west coast.

All of this has the consequence that the east coast does not have the frequent, large earthquakes that central and southern California does, nor does it have the string of active volcanoes that northern California north through southern British Columbia does.