Your source files are the vectors and they are the asset. Ask for SVG, which is what a website uses and what scales cleanly on any screen. Ask for EPS or PDF, which is what a printer, a sign maker, or an embroiderer will request. And ask for the original editable file from whatever the designer used, because that is the version anyone can modify later. Requesting these is not a courtesy. They are part of what you paid for, and a designer declining to supply them is the same problem a weak contractor agreement creates.
The raster exports are for convenience rather than permanence. PNG with a transparent background is the one that matters, because it sits on any color without a white box around it. JPG cannot hold transparency at all, which is why a logo supplied only as a JPG looks broken the moment it lands on a dark header or a colored slide. Ask for PNG at a few sizes, including something large enough for print at around two thousand pixels wide, plus a favicon file for the browser tab.
Formats are only half of it. You also need variations, and this is what people forget until they need one urgently. A full color version. A single color black version and a single color white version, for printing on merchandise or placing on a photograph. A horizontal arrangement and a stacked one, because the space you have will not always suit the shape you designed. And an icon or mark alone without the wordmark, for social profile pictures and app icons where a full logo becomes illegible.
Keep all of it in one folder in your cloud storage, organized by format, and store a copy alongside your business records rather than only in a folder shared with the designer. Add a short note recording the exact color values, in hex for screen and in the print equivalents, and the names of the fonts used. That note takes two minutes and it prevents the most common branding failure, which is not a bad logo but a business whose logo appears in four slightly different colors across its own materials.