Make client API calls relative instead of hardcoded localhost:5144

The client always called a hardcoded http://localhost:5144, which only
worked when browser and server shared the same "localhost" — breaks for
any real remote deployment, since the browser would try to reach that
port on the visitor's own machine instead of the actual server.

Now every environment routes /api/* to the server under the same origin
the page was loaded from, so the client code needs no per-environment URL:
- Docker (client container's own nginx) proxies /api/ to the server
  container.
- `npm run dev` (Vite) proxies /api to localhost:5144 via server.proxy.
- Production nginx (reverse proxy + TLS) just needs to forward everything
  to the client container, which already knows how to route /api itself.

Also incidentally removes CORS from the picture everywhere, since none of
these setups make a cross-origin request anymore.

vite.config.js

@@ -4,5 +4,11 @@ export default defineConfig({
   root: "src/Client",
   server: {
     port: 5173,
+    proxy: {
+      "/api": {
+        target: "http://localhost:5144",
+        changeOrigin: true,
+      },
+    },
   },
 });