It's not as simple as converting it; the formats requested by your client are vector formats.
What you use to draw/paint doesn't support those, which means you aren't using a vector art program.
Vector art is stored as actual vectors; each shape is stored with its coordinates and colors. What you produce, raster art, is a grid of pixels with a fixed size.
While vector art (ai, eps, svg) can easily be converted to raster art (jpg, png, gif), the other direction is much more problematic. Think of it like converting 3D scenery to a 2D representation (taking a picture, done) vs. converting a 2D one to a 3D one (manually creating 3D objects from a flat photograph by guesstimating what they originally looked like).
The process for creating vector art from pixels is called tracing, but the vectors it produces are usually not very clean.
I also have to wonder if your client is familiar with your work at all; if you exclusively do raster art, why would they request vector art from you? Maybe they *also* don't really know how those two formats are vastly different ways of storing 2D art, they've just heard that vector art is "better" because you can scale it to any arbitrary resolution without losing quality.
But requesting a hand drawn painting (as opposed to vector art) in vector format because you think that way you can scale it is utter nonsense.