If you know what a vector is, I'm impressed. In the "alphabet soup" of different file formats, there are only two types 1) raster and 2) vector. And raster files are by far the most common type of file - that's what a jpeg is (Photoshop). Vector files are created by programs like Adobe Illustrator, and are much rarer. And since most people have never even heard of vectors, explaining to a client that they should pay extra to have their logo done as a vector is quite a task! But it's well worthwhile.
A vector file is infinitely scalable. That means that you can make it as large as you want without it ever getting blurry, or getting "the jaggies". A vector file can be used for everything from a business card, to being displayed on the side of a building, with razor sharpness.
So, if you're client has heard of vectors, or they trust you to tell the truth, get them to invest in having their logo created, or re-created, in vector format. That will mean that they'll never, ever, get in trouble with sharpness and clarity however they use it, even on High-Definition TV.
The drawback, of course, is that it's difficult enough to talk techy to your clients (throwing around terms like eps, jpeg, psd, or pdf), but getting them to actually pay extra to have a file created in a format that they have probably never heard of, is nearly impossible.
I've been teaching vectors since the '90s. And even in classes that last a semester, with many assignments, the concept of vectors can be difficult for most of my students to understand. Once they use Adobe Illustrator, of course, they get it, but there really isn't much demand for vector manipulation programs among non-professionals, not like raster manipulation programs (which are programs to let you do adjustments to photos - which all phones have nowadays).
If your client is, for example, starting a restaurant, they're really going to want to get their logo into a vector format. If they started with something that someone did in Photoshop, or in an app, then it will probably work for them, for a while. But when they start making signs, or enlargements for posters, with their logo, it's going to look terrible, all blurry and jaggy.
And it gets worse. There's no way to just "click" and create a vector file from a raster. If it's very simple logo, with just a few simple shapes and some text, it can be recreated fairly quickly, but it will take hours, not seconds. And if it's complex, it will take several hours. And if you hire someone like me, and even if I give you the best price I can and discount it 50%, it will seem like a strange expense to a client who doesn't understand vectors.
I hope this helps. And yes, your client does need their logo recreated in vectors. If you can convince them, I'm ready to work.