Allowing sudo privileges, but not access to sites
By adjusting the sudoers file (/etc/sudoers) with sudo visudo it should be possible to accomplish this. It is a hell of a job and you need a profound knowledge of all commands to fine-tune this while avoiding errors and loopholes though. 

You have to add the user to the User privilege specification section

...
# User privilege specification
root    ALL=(ALL) ALL
%admin  ALL=(ALL) ALL
...


Then use a whitelist (or a blacklist) of allowed (or disallowed) commands:

Examples:

whitelist

# User privilege specification
root    ALL=(ALL) ALL
%admin  ALL=(ALL) ALL
user    ALL=/usr/bin/nano,/usr/bin/opensnoop


blacklist

# User privilege specification
root    ALL=(ALL) ALL
%admin  ALL=(ALL) ALL
user    ALL=!/usr/libexec/PlistBuddy,!/usr/bin/passwd,!/usr/sbin/*


You may mix whitelist and blacklist.

Please check man sudoers how to simplify things or narrow things down by configuring User, Runas, Host and Cmnd alias specifications.



Check the accepted answer to the question How to prevent sudo users from running specific commands? for the pitfalls of configuring a simple command like rnano in the sudoers file.