Are there any undergraduate research programs available online?
I think you should check The Open University's site. 

Under the Studying for a Research Degree section in the Faculty of Mathematics, Computing and Technology site I found this:


  Part-time research students pursue their projects at a distance, under individual guidance from expert supervisors.


So maybe it is a possibility if you actually would like to earn a degree from that university, but since this institute is famous for the excellent quality online and distance education I think you should write them a message and ask for information. I guess they would have good suggestions about your question.

PS. I've taken an online course from this University on the FutureLearn site, and the material's quality was very good.