You (and the screen printing house) are confusing two terms:
Vector: Artwork consisting of vector-based paths. One of the advantages of vector-based artwork is its "resolution independence." That means it can be scaled so that it can be rasterized (converted to an array of dots) at any size. That is NOT the issue in screen printing.
Line Art: Solid-color artwork. Artwork consisting of only solid colors. The opposite of continuous-tone artwork, which contains graduations of color. Line art can be printed as color separations which have only areas of black and areas of white; image area and non-image area. Printing color separations of con-tone artwork requires conversion of parts of the artwork to tiny dots to simulate the various tones.
When a screen printer tells you he "needs vector-based artwork" what he usually means (and should be saying) is that he needs color-separtated line art.
Both line art and con-tone art can be created in either a raster-based program (ex: Photoshop) or a vector-based program (ex: Illustrator).
JET