The reason raster formats are unsuitable for some applications is that they don't scale. A JPEG file is just a collection of pixels; if you want to blow it up on a large banner like one you can hang on the outside wall of a building, each pixel will be bigger than your hand. Something which looks like a circle at the original scale will look like increasingly sloping stairs because you start seeing the individual pixels when you blow it up enough.
Vector graphics, on the other hand, consist of drawing instructions which can be rendered exactly at any scale. A vector file consists of drawing instructions, like "draw a circle at this point with radius r" which can be scaled down up or down to suit the medium -- so if r is 25mm in the original drawing, you can scale it up to 25m or even 25km and still get a very detailed and crisp circle.
What ultimately makes sense depends on the use case, and the drawing. If it inherently displays something like a human face, or a heat map, or pixels which are supposed to look jaggy, vectorizing it won't really improve it (though you want to make sure the JPEG has enough resolution to look good at the scale you plan to use it). But if it's a typical logo, definitely vectorize it. Whether that can be done automatically or requires manual work depends on your ambition level and the complexity of the input picture. A simple logo with just a few colors can probably be done automatically with very good quality; a complex one will almost certainly require manual tweaks, or be redone in a vector drawing program.
(If the artwork was competently done, chances are it was originally created in a vector drawing program, and was delivered as a JPEG rendering by mistake, perhaps because of a human communications problem. Then of course just have the artist make it available in a better format.)