How can I join two solid surface vanity tops for a seamless appearance?
The makers of those tops will have the seam adhesive to join the tops permanently.
If they are anything like Corian by Dupont, they want to protect their warranty. You may need to do some convincing to an installer to sell you the adhesive you need. It can be done, I have done it before. It takes a router or a belt sander to get the joint down flush after the glue squeezes out.

More later if you go that route.

12-9-13 addendum

The bond is really strong, but it is always best to insure the joint is stable on both sides. This can be done with silicone under both halves. While silicone is still fresh, position your halves and use shims from the underside if needed. From the looks of it you may not have the luxury of getting a clamp block at the back edge but you may at the front. Have everything ready, and clamp the front and shim the back like you are ready to glue. Get it right at this point after this step there is no turning back. Use a straight edge of some sort to insure surface are flat and true. Remove all clamps that block access to the joint. apply your bonding cement and reset the clamps. The stick(s) I have referred to are small hardwood scraps, maybe 1/2" square to 3/4" square and 4 to 6 inches long. When this is placed on the glue joint it will spread the glue a little, but it will sand out with the rest of the joint.

After all this is said, any of this is for worst case scenario. If the joints line up perfectly as it sets on the cabinet, which it looks real good in the picture you offer, all you need is to apply the glue and let it set which will only take minutes to set, and hour to finish.