No, it's not possible to have them transparent. We could serve them as transparent .GIF images, but .GIFs only allow a pixel to be either fully transparent or fully opaque, which gives the pets a really ugly hard edge. We could serve them as transparent .PNG images, which allows semi-transparency, but these display with an ugly black background or black fringes on Internet Explorer 6, which is still a popular browser.
Here's our image with a background colour behind it:

The same image on a dark background:

Here's a transparent GIF version (hard edges), here the transparency is diffused to attempt to fake semi-transparency, with a white matte:
hardedge.png
The same image displayed on a dark webpage:
hardedgedark.png
Here's a transparent GIF version (hard edges), the transparency is not diffused, again with a white matte:
hardedge2.png
The same image displayed on a dark webpage:
hardedge2dark.png
Here's a transparent GIF (hard edges) with no matte colour, fine details like balloon stalks and whiskers disappear:
nomatte.png
The same image on a dark webpage:
nomattedark.png
And here's a transparent PNG on IE 6 (who knows how it chooses the weird background colour):
ie6.png
(Transparent PNG works perfectly on Firefox and IE 7 and upwards)
The effect is less extreme on many of our pets, but the way we have it set up now allows us to be much more flexible with our artwork. If IE 6 could display them correctly (and particularly on forums where we cannot use any special HTML code to work around IE 6 problems), we'd definitely use transparent .PNGs.