Drag and drop has never worked very well in QtWebAssembly.
But it is working in 6.6.0beta1 single threaded release.
You need to use asyncify though:
Good news is, you no longer have to rebuild Qt to get asyncify, you just need to add a linker argument.
cmake: target_link_options(<target> PUBLIC -sASYNCIFY -Os)
qmake: QMAKE_LFLAGS += -sASYNCIFY -Os