wasm-git (and the libgit2 library it is built around) use the GPLv2 with linking exception. This exception reads:
In addition to the permissions in the GNU General Public License,
the authors give you unlimited permission to link the compiled
version of this library into combinations with other programs,
and to distribute those combinations without any restriction
coming from the use of this file. (The General Public License
restrictions do apply in other respects; for example, they cover
modification of the file, and distribution when not linked into
a combined executable.)
Normally, using GPL-covered code in a webapp frontend would trigger the full breadth of GPL license conditions, such as offering the complete corresponding source code of your frontend under a GPL-compatible license. This is clearly a kind of distribution or copying, especially since recipients would be able to make further copies from the file they downloaded from your servers. It is clearly not distribution if the wasm-git files are only used on your backend.
But due to the linking exception, you can integrate an unmodified wasm-git “binary” into your frontend, without extra license obligations. This linking exception is more permissive than e.g. the LGPL. But as I read the linking exception, you should still offer the complete corresponding source code for wasm-git and libgit2 to recipients of the software, and provide them a copy of the license. The code you write is unaffected by the license, though.
Do read the libgit2 license carefully since it includes a lot of bundled software, including some under the LGPL (which is similar to the linking exception, but requires you to enable users to use the software with a modified version of the LGPL-covered code – tricky in a web context due to CORS).