What kind of camera and computer equipment would I need to create "old
fashioned" photos (e.g., "tintype," or with a brownish-tinge finish? 
I am interested in starting a business that would require the
appropriate technical equipment as well as costumes.  Where might I
find these?

Hello.

Antique and Amusement Photographers International (AAPI) has a web
site devoted to the "old-fashioned" photography business. Their web
site is at Oldtimephotos.org:
http://www.oldtimephotos.org/

AAPI lists a variety of vendors that deal with equipment and costumes
for this type of business. See AAPI: Vendors
http://www.oldtimephotos.org/LocatorMaps/LocatorMapVendor.html

If you're going to start this type of business, you might consider
joining AAPI.
AAPI: Membership application
http://www.oldtimephotos.org/html/page2b.html

Basically, there are several different options for this type of
photography (e.g., digital camera with a sepia filter, regular film
camera with a sepia filter, or polaroid camera with sepia film).

Ashley Nostalgia is a company that specializes in selling camera
equipment and costumes to "old time" photo business. On their web
site, they mention special software and equipment that they sell.
See Nostalgiasite.com for more information:
http://www.nostalgiasite.com/index.html

Some "old time" photographers use Polaroid cameras and a special
Polaroid film (606 Sepia) made for "old time photography."
See this data sheet from Polaroid:
http://www.polaroid.com/service/filmdatasheets/3_4/606fds.pdf
(The document is in PDF format, so the Adobe Acrobat Reader is   
required. If you don't have that, visit:
http://www.adobe.com/products/acrobat/readstep2.html )

If use a non-Polaroid camera, a Sepia filter is often used to get that
"old-fashioned" look. BHphotovideo.com sells such a filter made by
Cokin:
http://www.bhphotovideo.com/product/168444/COXP005/REG/4033

As for costumes, there are several dealers that specialize in selling
costumes to "old-fashioned" photographers.

Profbloodgood.com is a costume dealer  for the "old time" photo
business.
http://www.profbloodgood.com/
View their costume catalog here:
http://www.profbloodgood.com/shopping/enter.html

PCostume.com also makes costumes for this type of business.
http://pcostume.com/oldtime/index.html

Ashley Nostalgia's costume catalog may be accessed via this link:
http://www.nostalgiasite.com/htmcat/start.htm

You might also find this newsgroup discussion interesting:
"'old-timey' vintage-look costume photos," hosted by Google Groups
http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&threadm=bruce.894.053B7F5D%40plugin.com&rnum=1&prev=/groups%3Fhl%3Den%26lr%3D%26ie%3DUTF-8%26selm%3Dbruce.894.053B7F5D%2540plugin.com

search strategy: "antique photo", old-time photos, cameras, sepia,
polaroid, filter, costumes


I hope this helps.