First try if you can ping the cloud server. If you can ping, then you have a path to the server from your new computer. That does not necessarily mean that you can connect to the server. If you don't get a ping, that does not necessarily mean you don't have a path to the server because some sites disable ping responses.
Find the networking guy or girl in your organization, go over to their desk and ask them if there are any firewall rules preventing you from accessing the outside world and specifically this server, specifically from your new computer. This should be a piece of cake for them to look up. If there are firewall rules that prevent you from accessing that site, have them change the rules to allow it.
If there are no firewall rules on your side that is preventing access, talk to the vendor and ask them if they have specific white-listed IP addresses that are allowed to access their server. It could be that when you replaced the computer, you are accessing the remote server from a new IP address and the vendor has not white-listed this new address.