Earlier quoted context omitted.
That's not how it works. Shodan continuously checks all IPs on the Internet. When you visit the website you're checking for existing information in the database. Your use of the Shodan website/ API doesn't change how Shodan crawls the Internet.
How do I check my server, which doesn't have a browser installed?
ssh -D9999 yourserver
then set up the SOCKS proxy in your browser's settings