A vector image in the eps file format is the gold standard for all logo files. All eps images are vector-based, when exported from Adobe Illustrator, they're meant for print usage. This means an eps file can be sized up or down without deteriorating the image quality. You can get any png, jpg, or gif logo converted to an eps or svg vector image at places like Fiverr for about $20. There's also free converter tools on the web, but they will only do one color for the entire image. To do multiple colors you'll need to hire someone to trace the logo or build it from scratch in vector and add colors.
As for a website, it's best to use a svg file because it's much smaller than an eps file, and svg has almost the same quality as an eps file on websites in all browsers.