I had the same issue and couldn’t get it to work even after installing GDAL, Fiona binaries, etc. I finally managed to get it to work by building Fiona from source.
I installed GDAL using the OSGeo installer (https://live.osgeo.org/en/win_installers.html). Then after cloning the Fiona repository you need to pass the locations of the GDAL include and lib folders, as well as the installed GDAL version. For me this was:
python setup.py build_ext -I "C:\OSGeo4W64\include" -lgdal_i -L "C:\OSGeo4W64\lib" install --gdalversion 2.2.4
Note that I only tested installing fastai via running python setup.py install. However, I was getting the same error using either installation method yesterday.