Can an onboard laser propel a solar sail?
In theory, a tiny, tiny, tiny little bit. In practice, no. You'd be better off without the sail and pointing the laser backwards, but you'd be even much better off without the laser.
Warning: sarcasm follows.
A one point twenty-one gigawatt laser would provide a paltry four newtons of thrust. There is no such thing as a continuous wave gigawatt laser; we'll need a thousand or so one point twenty-one megawatt lasers (they don't exist, either). Lasers aren't exactly efficient; I'll be very generous and assume a 25% efficiency. All we need is four Mr. Fusions and voila! our thousand laser farm has power. Being even more generous and assuming each of our non-existent megawatt lasers weighs a metric ton, that's an acceleration of 4 microns/second2. (That's a microscopically tiny acceleration.)
Since Mr. Fusion doesn't exist, we need some other power source. Mr. Fusion was nice and compact. The world's largest nuclear reactor, the Kashiwazaki-Kariwa Nuclear Power Plant, was rated at 8.21 gigawatts, two more than we need. Since Japan no longer wants their nuclear reactors, all we need to do is package that plant for use in space. One minor problem: The extreme mass will make our already paltry 4 microns per second2 orders of magnitude smaller yet.
Since Mr. Fusion doesn't exist, and nuclear reactors are a lousy power source for a space-based laser, perhaps solar arrays (very, very big solar arrays) might work? The answer is no, they won't. Now we're stuck with the exact same problem that makes solar sails rather useless after five AU or so.
Bottom line: You'd be much better off without the laser.
What about the idea raised in Prospective of Photon Propulsion for Interstellar Flight, which was pointed out in a comment to the question?
Note very well: This laser is not mounted on the vehicle. Even the author of this conference paper knows that mounting the laser on the spacecraft is sheer nuttiness. What he doesn't know is that his idea is sheer nuttiness, too.
His idea is based on the 2001 conference paper Metzger and Landis (2001) "Multi-bounce laser-based sails," Space Technology and Applications International Forum-2001, Vol. 552:1. This paper has been cited a whole seven times since 2001, with six of those citation coming from the author of the paper in question.
The basic idea is to have a fixed laser source such as on the Moon and have a mirror at the laser source and a mirror on the spacecraft. The laser hits the spacecraft mirror, bounces straight back to the source mirror, which bounces straight back to the spacecraft mirror, and so on, a thousand times over.
This idea is completely divorced from reality. This places pointing accuracy requirements on the spacecraft that makes the pointing of the Hubble Space Telescope look absolutely amateurish. This places flatness and reflectivity requirements on the mirrors that are far, far beyond our dreams of Mr. Fusion.
Finally, if by some miracle this idea does work, do you really want to be on the receiving end of the equivalent of one point twenty one terawatts of laser power?
