Do Business proposal use quantitative or qualitative research methods?

Business proposals should be done using both quant/qual. The qualitative should be done first to give you and idea about potential solutions to the problem you're trying to solve or the opportunity you're addressing. This data will not be of a statistically significant nature, and should not be taken as representative of the population. Based on the outcomes of the qualitative, you can then conduct quantitative to "prove" the resulting hypothesis (es).

Based on the results of the quantitative, you then have a foundation on which to build the rest of your biz prop, ie: projections, financials, etc.