You can set up a virtual machine and give it (almost) whatever IP address you want, but it won't help you at all.
10.x.x.x and 192.168.x.x are both ranges of private addresses. These addresses are only used inside an organization, they are not routed on the Internet.
Amazon doesn't see your work machine as 10.x.x.x, it sees it as coming from some gateway machine that sits between your work machine and the Internet. The gateway machine performs NAT to enable the work machine with a private address to talk with the outside world — without a public address, a machine cannot talk directly to the outside world.
If you try to connect directly to the Amazon machine from home, you connection will appear as coming from an address that belongs to your ISP at home. If you use the VPN, your connection will appear to come from the Internet gateway for that VPN. In typical corporate VPN setups, that would be the same gateway as when you are at work. If you can connect to Amazon from work but not when logged into your corporate VPN, then it's due to something in your work's network setup — presumably there's a policy that forbids this.
If you can't get the policy changed, you may be able to use another work machine as a relay. Let's say the Amazon machine's address is 203.0.113.65, that you can log in to a machine 10.1.2.3 in your corporate environment over SSH, and that this machine is allowed to talk to the Amazon machine. You can make 10.1.2.3 relay your connection to the Amazon machine from your laptop at home:
ssh -L 1234:203.0.113.65:1234 10.1.2.3
Then tell your software on your laptop to connect to localhost:1234. The ssh client will relay the connection to 10.1.2.3 which will in turn relay it to 203.0.113.65.