Lets say that the US Government had a subpoena to get logs of user activity. However, assuming the VPN provider was not previously logging, could the government ask the provider to start logging, in hopes that the user who the subpoena was intended for would continue to use the VPN?
Yes, in a number of ways.
But, a local prosecutor could simply ask an ISP or VPN provider to log a user. Prosecutors have a lot of power and discretion in terms of enforcing the law, and an ISP or VPN provider may simply not want to be forced into action by a subpoena.
A local court or grand jury can subpoena the ISP or VPN provider to start logging a suspect's activity. A subpoena requires a certain amount of evidence of illegal activity to convince a court or a grand jury to force a VPN provider or ISP to log a user, but it is, of course possible. As a side note, the existence of such a subpoena could be public, but could be sealed by the court. The VPN provider or ISP will be legally obligated to obey the subpoena and start logging; if not, they could be forced by a court order into compliance, or charged with contempt if they don't comply.
A federal government law enforcement entity - like the DOJ - can subpoena (9-13.000 - Obtaining Evidence | Department of Justice) a company to start logging a user.
Depending on the nature of the suspect and their activity and international nature of the traffic, the DOJ could get a United States Foreign Intelligence Surveillance Court (Wikipedia) warrant for logging and surveillance. This will also require evidence and the sign off by a number of judges; the VPN or ISP will not be able to disclose to the public or the target individual that they are complying with such a subpoena.
At the most extreme in terms of surveillance and secrecy and which could be argued is extra-judicial, the NSA could simply use a system like PRISM (surveillance program) (Wikipedia) and not need a subpoena or issue any notice to the ISP or VPN that they are logging users. The ISP or VPN provider may not even know it is happening.
Or the NSA or similar government entity could another signal intelligence system we don't know anything about. The other four of the Five Eyes (Wikipedia) could even be involved.