To expand a little more on my post....This is an excellent idea. In fact, it could probably be done for not a whole lot. The entry-level SpaceX rocket costs $6million per launch, and NASA could probably get a (very) bulk discount, so lets say $5m. If they launch 50 times a year (take a couple weeks off for christmas or something), that's only $250million...just chump change in NASA's budget.

Then it would be an issue of finding payloads for these 50 launches each year. Ideally NASA could have a competitive process that would let individuals or organizations compete for the free launches. Then if the org isn't able to get their payload ready in time, you could have a pool of ready-to-launch payloads that didn't win the competition, but are willing to launch if there is a chance. If all that fails, the rocket should launch anyway, and just do a sub-orbital flight, so that the system stays sharp.

There's a couple issues with this...one if the biggest is range tracking and availability. Having to be available to launch every week would be very tough for the range, because it usually takes them 24hrs+ to get set for a launch, then there are several days booked where the launch can happen any time in that window. To solve this problem, something that launches with this regularity would need to be independent of the current assets, which with cheap GPS and sat data links shouldn't be too big of a deal.

Another thing to think about would be orbital debris. Launch every week, with the competitively determined payloads, which probably won't be as long lived as traditional payloads, could contribute negatively to the amount of debris in orbit. To mitigate this, a big part of the competitive selection process should be how the payload will de-orbit itself, even in the event of a failure.