Cannot access internet from VMWARE ESXI host connected to an Internet Router

You need to set static IPs for these VMs in their network settings unless you want them to pick up an address from the router, in which case you just need to make sure that you have these VMs on the same network (vSwitch) as the host.

And by static I mean you need to set the DNS & DHCP server address as the VM that is hosting those roles.  So just make sure that their IPs are on the same subnet as the router, with it as the gateway and then DNS.  

If you want to pick up a DHCP address just leave the adapter as it is and it will pick up a DHCP address from the server hosting that role.

Lastly, make sure in DHCP options that you've set the default gateway so that the computers that pick up a lease from DHCP will get that set for them.

If that doesn't make sense, please reply and I'll help you.

You may want to think hard, about adding Virtual Machines, with Windows OS directly on the internet with Public facing IP Addresses, and use firewalls or a traditional NAT network to protect your internal virtual machines and their network, using a virtual router.

@Andrew Hancock This is just a test environment. I understand that I need to have these closed off via a firewall but I am trying to download critical files from the internet to set up a test build. I am happy if it is just one PC that has access.

@Brad, all my internal VMs have a static address and IP ranges + Subnet masks in the same range as the router. I have even set the router's IP as the gateway for these. Am I missing something?

in that case, you will need to connect a virtual machine network interface, to a virtual machine VM Portgroup, on the same virtual switch as the Management Interface of the ESXi server, and allocate a static IP Address, or DHCP.

Do you have more than one public facing IP Address?

@Brad, all my internal VMs have a static address and IP ranges + Subnet masks in the same range as the router. I have even set the router's IP as the gateway for these. Am I missing something?

The real question is this:  Can you get to a website by IP from these VMs?  Try going to 74.125.228.35 in your web browser from one of the VMs.  If it takes you to Google.com then it's definitely a DNS issue and we can go down that road.  If that doesn't work, try opening a command prompt and pinging 74.125.228.35.  Let me know if you get replies.

Can this be done even though it is sitting on a dynamic IP address?

DHCP may or may not issue a Default Gateway.

If no Default Gateway is provided, you will need to issue static IP Address with Default Gateway.

Got my answer based on that response.