Did you manually forward the ports to your machine as the error says?
No VPN-provider will help because you can't port forward, even those that let you it's usually a random port but Warframe needs specific public ports, I tried forwarding those random ports to 4950 & 4955 but warframe didn't use them. If you can't portforward on your regular connection you can rent a vps and ip, setup a vpn server, and portforward that to your machine, but keep in mind you'll probably pay for traffic so no youtube while playing.
Other than that there is no way around it unless DE buys some servers for normal multiplayer which they won't do, you see how much they care about their thread and enough payers don't seem to care they are being used as infrastructure. Anyway, to check whether your port is open you can let iperf listen on that port and if you can connect from the outside, eg. mobile app not in your wifi, it's open. You'll have to specify udp iirc iperf defaults to tcp.