do you want to know how to connect your local database to your vercel deployment?

if so that depends how your local database is set up and served

i don't think this is particularly secure or even a good idea, but if your local database isn't explicitly available from the deployment, you'll want to expose it to the internet somehow via something like tailscale funnel, ngrok, or cloudflare tunnel. you can point the deployment to the url it gives you and it should work.
though, i would highly recommend having the database on a hosted database management service or on a vps or similar with backups enabled to offsite

I will try to use ngrok , i wanted to use a local database just to study how it works, and maybe have it as a long term solution . Have you tried to make your on database without host.