Finding watershed area for each point in river network using DEM using QGIS

You will definitely need an elevation raster (DEM) to do what you want. With the DEM you can use two GRASS modules to get individual drainage areas for points along the river network as follows: First calculate a flow direction grid from the DEM with r.watershed:

r.watershed elev=DEM thresh=<your threshold> drain=flow_dir_grid

See the r.watershed manual for more details Next loop thru the X-Y values of the points along the river network to calculate each watershed with the r.water.outlet module.

r.water.outlet drain=flow_dir_grid basin=basin1 easting=X1 northing=Y1

Details in the r.water.outlet manu page

For doing the loop, you might be able to get some ideas in this post

r.stream.basins would do what you are after. https://grass.osgeo.org/grass70/manuals/addons/r.stream.basins.html

But since it is (currently) a GRASS addon, it is not available in QGIS. It might be added as a core module to later versions of GRASS...