Client-side JavaScript cannot access MySQL without some kind of bridge. But the above bold statements that JavaScript is just a client-side language are incorrect — JavaScript can run client-side and server-side, as with Node.js.
Node.js can access MySQL through something like https://github.com/sidorares/node-mysql2
You might also develop something using Socket.IO
Did you mean to ask whether a client-side JS app can access MySQL? I am not sure if such libraries exist, but they are possible.
EDIT: Since writing, we now have MySQL Cluster:
The MySQL Cluster JavaScript Driver for Node.js is just what it sounds like it is – it’s a connector that can be called directly from your JavaScript code to read and write your data. As it accesses the data nodes directly, there is no extra latency from passing through a MySQL Server and need to convert from JavaScript code//objects into SQL operations. If for some reason, you’d prefer it to pass through a MySQL Server (for example if you’re storing tables in InnoDB) then that can be configured.
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JSDB offers a JS interface to DBs.
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A curated set of DB packages for Node.js from sindresorhus.