Hi johny.vent,
It looks like you are trying to access to an external application from your VM, In that case that Application need to add your Public IP of the VM to the allowed consumers (whitelist). If your question is about change the IP public for one IP that is in the allowed range of you application , as I know that's not possible because Azure Manage their own specific Public IPs address pools.
So the best option is get on your virtual machine your existing ip Public( ifconfig.me ) or Azure portal check which IP public do you have. After that request to your application owner to white list that IP address.
Let me know if this comment help you to solve your problem.
Luis