I'm flabbergasted that one of the other meta-packages didn't pull it in, but try installing the libqt6webenginecore6-bin package. It is literally a package that supplies ONE binary file. Which is QtWebEngineProcess! That's nuts! It's all about debundling these days.
The good news is that you shouldn't have to rebuild Sigil. Just install that package and try running Sigil again.
It's no wonder there's no Qt6 version of Sigil in the Ubuntu repos yet. All the Qt6 pieces are there, but no one can figure out which of 60 or so Qt6 dev pieces (and/or lib packages) are needed to build Sigil. Let alone figure out which tiny qt6 tool/bin packages might be needed to run it!