Yes, you also can write a function that the AI can call with an internet search query, and then fulfill it by an external API to get search results. Then another function for getting web pages, even ignoring robot files meant for OpenAI and not “meant” for you.
Just as you can write a function to let the AI multiply two numbers reliably.
See the Bing API price per query: https://www.microsoft.com/en-us/bing/apis/pricing