As above, really; I used to do this all the time years ago where the firm was supplied with printed logos and we needed vector versions for digital output from an image setter. I’d scan the logo and import it into CorelDraw, later Illustrator after moving from PC to Mac, then draw over the top and bin the bitmap when finished. That way any colour can be used to fill inside the lines.
I did use Autotrace, but it tended to put hundreds of extra nodes in, and make straight lines and curves all wonky, needing lots of work to clean up.
Trace in illustrator is much better now, but I’m really not sure if there’s a cheap or free equivalent. CorelDraw is very much more intuitive to use than Illustrator, and is what I’d recommend, but I have no idea about costs these days, it’s years since I used it.
There’s a real knack to placing nodes, then dragging handles to get the vector lines to follow the underlying logo correctly, but once you get it, it’s brilliant fun, I used to love re-drawing logos, did hundreds of them. Best job I ever had. 🙁