The Makefile is written for a linux environment.
To build libhdf5, you must have Emscripten installed and activated, so that the emcmake and emmake and emcc commands are available.
Then just type
make
libhdf5.a and libhdf5_hl.a are built in ./wasm_build/hdf5/lib/
and are used to compile ./src/hdf5_util.cc to ./dist/esm/hdf5_util.js and ./dist/esm/hdf5_util.wasm
which are then used by ./dist/hdf5_hl.js
(and also ./dist/node/hdf5_util.wasm which is used by ./dist/hdf5_hl_node.js, the nodejs entry point)