@Sebastien_Jourdain is correct – you can use the vtk.js ITKImageReader to read in a volume in VTK legacy file format (binary or ascii) via itk.js.
As a side note, it is now also possible to read it a vtk.js PolyData from a binary or ascii .vtk polydata or unstructured grid file via VTK C++ WebAssembly / itk.js with the vtk.js ITKPolyDataReader.