Is an EPS file, a vector file?
Not necessarily. An eps file may contain raster, vector, or both. Both vector and non-vector programs can save as eps.
EPS stands for Encapsulated PostScript. EPS is a graphic file format that describes an image (raster/bitmap) or a drawing (vector) or both. It is a standard file format for importing and exporting PostScript files. Wikipedia
True vector graphics are created with vector tools. Even a file created with a vector program may not be 100% vector. If you create a file using a vector program but you import, place, or paste a raster (bitmap) element - like a photograph or a jpeg- into your vector file, you then end up with a file that has both vector and raster elements. Also, some of the effects within vector programs like gradients and shadows are actually raster effects. When a company requests for 100% vector art that means you cannot have any raster elements in your vector file. All elements within the vector file need to be created using the tools of the vector program (like the pen tool); no placed, imported, or pasted bitmap elements allowed and no raster effects. Raster vs Vector
Some of the processes that must have 100% vector art are sign vinyl-cutting and engraving. The equipment these processes use must follow the vector outline/wireframe.
Here at Vector Conversions we use Adobe Illustrator and supply you with 100% vector art unless requested otherwise.
Which programs/applications are vector?
Here is a list of vector graphics editors: Wikipedia