For a whole mat in a flat color- tricky. Hard to get an even coverage. Also it would make the mat
curl up. Also also, it would stick to the glass and/or make annoying optical artefacts. All-in-all a no go.
Using thin washes and sponging/stippling works better. E.G. if you use an Ingres paper faced board in a muted
tone and stipple it with a thin coat of acrylic in the same general color but darker, you can do a very convincing
velvet with none of the hairs coming off. Only really applicable if you want that sort of finish.
For reveals on double mats it's a more practical proposition. Use acrylic paint and apply at least two coats.
Ditto what Prospero says: If the customer is so finicky that you cannot find the right colour in all the matts available they will probably not be happy with whatever paint job you can offer either. I am guessing that the time you have already spent with this customer is putting the job close to non-profit status?
What Mar said. One of the fun things about doing this is that you can be as messy as you want on the surface, and when you lay the top mat down it looks perfect. This is a good opportunity to do some faux texturing or other special effects, which are really easy but look extra-special. Since the cost factors are low, you could surprise your customer with a "free upgrade".
I have been successfully coloring bottom mats with acrylic paint for years. But follow Framar's advice and paint the mat before cutting. I use one of those small rollers that is 2-3 inches wide and paint a swath.
Acrylic paint should not compromise the art, but if you are concerned you can use a reverse beveled undermat. That brings the surface of the paper to 1/8 from the art. I have never had problems with curling, but if you were painting the whole surface, I would paint the back too.
And if you want to go down a preservation rabbit hole, start asking suppliers about the composition of the top papers on colored mats.
There was a time when there were 2 colors of matboard. White and white. Framers used airbrushes (they were called something else) to paint them to the color desired. I have often thought it might be fun to experiment with this process since it should allow for very thin but even coats.
Wizard experimented with a process to print colors on a matboard a few years ago, but I don't think it was successful or was too expensive to pursue. I never heard more after the original discussion.
In rare cases, I have been able to print a solid color on Epson single-weight matte paper, then drymount it to a white matboard to get the look I wanted. This is also a very successful way of producing patterns or images. But it is also an expensive method since it uses a great deal of expensive ink.