Since the federal government is currently only funding a very limited number of embryonic stem cell lines, I was thinking of voting with my pocketbook (so to speak) and throwing a few bucks at an organization that will more broadly explore the potentials of using stem cells from human embryos.
Is there any organization I can contribute towards that does this?
There is the recently-formed Harvard Stem Cell Institute, but they seem to lack their own web page so I can’t find information about donations. 
The problem is fairly complicated. Even if researchers get funding from a private source, they cannot use equipment paid for by NIH dollars (say, from earlier research) to do the work. They have to set up a brand new lab. So it’s expensive, and those private dollars don’t go as far.
I believe the Juvenile Diabetes foundation is funding stem cell research privately. I’m guessing a number of other disease-based foundations may be doing the same; I guess that would take some digging around. Harvard University has a lab that is completely NIH-restriction free with private donors, and they’ve sent out a number of stem cell samples to others who have the labs and funding that will allow them to do the research.
You might also consider putting your money towards advocacy, or behind backing certain state measures (like California’s Prop 71) which wants to sell bonds to fund this kind of research, bypassing the feds.
The OP is about charitably-funded organizations. I’d like to ask also if there are any private enterprises in the business?
60 minutes did a piece on the H. Hughes Medical Institute.
I think they mentioned that they do stem cell research
I dunno if they take donations (they are doing VERY well moneywise)
Brian
Off the top of my head, Geron Corporation - GERN, and Stemcells, inc - STEM. All the usual caveats about highly speculative biotech stocks apply here.