Though I'm not an expert, when it comes to sewing a particular fabric, I don't think the machine itself makes much difference. Assuming you've got a basic, sturdy machine, and not a kiddie version, it should be able to handle most fabrics, from silk to upholstery. Having said that, I have very little experience sewing through faux fur, so I couldn't tell you what setting you should have it on or what needle type to buy (sorry). One tip I have learned, though, is to *always* put something like tissue paper (that can easily be torn away from the stitches) between the fabric and the machine if you're sewing anything that has long, loose fibers. Otherwise you'll spend forever trying to untangle fur from your bobbin case.
Is it important that you be able to buy one online? I found my current machine through craigslist (I think there's a craigslist.co), and was able to get an older and much better model than I could otherwise have afforded for a really good price. As long as you don't need it right away, it might pay to shop around at websites like that first.
Sorry I'm not more helpful, best of luck with your search!
P. S., Your monsters are absofreakinglutely adorable! ^.^