No reason you can’t build a PHP app with OpenAI APIs, the APIs are very simple. Any cheap VPS host is enough to get started (Linode, DigitalOcean, etc). I’d imagine GoDaddy and HostGator would also work (though maybe not their super limited website builder products, but anything that can run PHP should be fine to start). It’s not a one-click install, but if you’re comfortable writing modern PHP it’s easy to get started. Note that you don’t “host OpenAI”, they provide standard HTTP API endpoints to use GPT, so as long as the host supports PHP you’re probably good.
My preferred PHP OpenAI library (not affiliated):