I own and operate a website pretty much on my own. I have a few
thousand stickers that I would like to offer to my visitors for free or for a
small cost. I do not want to deal with the payment (if I do charge)
and mailing process though and was hoping to find a company to do this
for me.

I'd imagine there are quite a few companies that offer this kind of
service, but the problem is that the volume wouldn't be that large and
neither would the cash flow (not doing it to make a profit) and I don't
want to pay too much for the service.

Are there companies that can handle this for me at a reasonable cost?

Thanks!

Note: I'd like to go the Cafepress route, but they unfortunately don't
do diecut (cut to shape) stickers and the stickers would be fugly if
they weren't. A company similar to them where I provide the stickers
is EXACTLY what I am looking for.

In order to easily find a reputable and affordable provider, I would
suggest you visit the following website:
http://www.elance.com/c/static/main/displayhtml.pl?file=eol.html&module=home

This website is similar to ebay in that you post a detailed job
description (in this case for mailing stickers), and people
competitively bid to perform the service.  You decide who wins the bid
based on their price and reputation (you aren't forced to choose the
lowest bidder if you don't trust them). Alternatively, you can review
a listing of around 1,000 bulk mailing providers by doing a search for
"mailing" or "bulk mail", whichever is more appropriate.  Again in
this instance, you can see how people have rated the providing
companies and how much business they have done through the site.

For more information about elance, you can review:
http://www.elance.com/c/static/main/displayhtml.pl?&file=seller_faqs.html

Hope this saves you time, effort, and money.  Good luck!