Are there "gig economy" resources in academic research? Are they fundable?
I've seen this happen in a sort of temporary 'research assistant/intern' role. Normally there's data that needs to be crunched or transcriptions done and it works to have someone come in to do it whilst the main researchers are doing the actual analysis/research. These can be paid, but usually tend to be from a local university or similar. 

For bigger projects, there are transcription and data services available online which can do large volumes of that sort of work for you, normally on a per page/per data set model.

I know you have explicitly said you don't want opinions on if it's a good idea, however I can't ever imagine many people wanting to farm parts of their research out to unknown people on the internet. The vast majority of research is sensitive either in terms of the raw data or in terms of the research goal. It would be too easy for someone to copy your work and beat you to the punch if you were working on something of interest. Therefore I can't really say how it might be funded or what it might be  called, because I can't imagine it actually coming into fruition