It's not possible to host the API to vercel. Vercel doesn't support laravel API deployment. So you've to host the API on a separate server. We suggest you create a VPS server and host the API there.

To host the API to a seperate server, follow this doc from Beginning to Install API

After host the API successfully follow this part,

Frontend

Now for frontend add API URL and other necessary config details to,

shop -> vercel.json

Open shop -> next.config.js and add your domain to images object

If your API is hosted on a subdomain, then add that subdomain with root domain on next.config.js

admin -> vercel.json

Open admin -> next.config.js and add your domain to images object

If your API is hosted on a subdomain, then add that subdomain with root domain on next.config.js

after that, install vercel cli on your computer using this command,

npm i -g vercel

After that, log in to vercel using this command,

vercel login

Then go to the shop directory and use this command to deploy,

vercel

Similarly, go to the admin directory and use this command to deploy,

vercel

For more details,

Please follow nextjs deployment docs: