How to find out the type of Internet blocking and bypass it?

Question

Ping

I work as a system administrator, I ran into a problem, they asked me to check the reliability of blocking the Internet on a working machine, the machine, in theory, has access through a local network, there are suspicions of a local proxy.
- ping vk.com, mail.ru - free, there is an exchange of packets
- the hosts file on the local machine is clean
- pages do not open in the browser.
- the window for changing network adapter settings is blocked, most likely by group policy.
tell me what kind of blocking this case can look like, and how you can find a hole in it.

Answer 1

ping vk.com, mail.ru - free, there is an exchange of packets

What will the user benefit from ping VKontaktika? Yes, absolutely nothing :) Checking does not make sense.
In a corporate environment with blocked adapter settings, it makes no sense - there are no admin rights, the hosts file is not corrected.
Proxy settings open? Proxy installed?
If a proxy is configured, if it has lists of restrictions - then depending on these lists and the paranoia of those who maintain and manage them - either a plugin with a VPN out, or nothing at all :)

Answer 2

There are many ways to block resources and many ways to manipulate ping/traceroutes. Offhand:
and more to come
Run netcat to the target hosts from the local machine to the required ports. You will see everything.