Fake fur generally can't be dyed. It's usually made from synthetic fibers which won't take dye (only natural fibers like cotton can be dyed). Some sort of industrial dyes will work but it's not a process you can do at home. I am sure there are exceptions to this, but that's generally how it works.
When we have to have a custom-dyed shag fur we use mohair or angora hides (about $150 - $200 Cdn per 3/4 yard). The best supplier we have found is Orappa Creek in Alberta Canada. They deal with a lot of puppet builders, doll makers and artists and they are very nice folks. Their contact info is:
Orappa Creek
RR1
Pickardville AB
T0G 1W0 Canada
1-877-5MOHAIR
http://www.orappacreek.com
orappa@telusplanet.com
*Ask for Helen or Don Gladson
The more expensive route would be one of the aforementioned custom fur manufacturers like National Fiber Technologies. Their info is as follows:
National Hair Technologies, LTD
300 Canal Street
Lawrence, MA 01840
1 800 842-2751
Other US fur manufacturers I know of that *might* do custom work are:
Monterey Mills
(800) 432-9959 or (608)-754-2866
PO BOX 271
1725 E. Delavan Dr.
Janesville, WI 53547
TexTenn
1 800 251-3027
ask for Katy Seifert
Big 4 Pile (212) 354-4155
Model Textiles - (973) 779-0276
If you don't have tons of money to spend I'd suggest searching ebay (Toasterboy is right - there's tons on ebay), the net and local fabric stores for furs. Rarely have we ever needed a fur that we couldn't find (but sometimes it takes
alot of searching). Another low cost option might be to experiment with wool.
Good luck!