I am considering getting a new front derailleur. Since the derailleur is riveted together so far as I can tell, it looks I will need to break the chain. The chain doesn't have a master link, so I'll need to use a special tool to push one of the pins through. After I do this, will I be able to secure the chain back together after I install the new derailleur?
Fifteen years ago the answer was a simple "Yes". Today's 10, 11, and 12 speed chains are not designed to be reassembled this way, and in fact, will reliably break or fail where you reuse the rivet. For these new narrow chains, simply use a replacement rivet on Shimano, or master link on SRAM or other.
For older bikes, or chains designed for 9 or fewer rear cogs, see below:
The same tool that you use to break the chain can be used to push the rivet back in. The trick is to make sure that you don't pop the rivet all the way out--it needs to stay in one of the sideplates. A lot of chain tools will stop automatically before they pop the rivet all the way out.
Alternatively, you can remove the rivet entirely, and the outer half-link, and replace that with a master link. You can get individual master links. These need to be sized to your chain (so if you have a 9-speed drivetrain, you'll have a 9-speed chain, and need a 9-speed master link).
For reasons of basic longevity of your chain split tool, I recommend you use a grinder to grind the head off the pins, right the way down to the face of the link. This will make the pin far easier to press out and less likely to ruin your chain splitter by bending the drive pin. Pins are usually retained by peening - flattening the protruding end so that [parts of] it become[s] wider, like a mushroom head. While you can force this wider part of the pin back through the smaller hole in the link, if the peening has made it dome shaped you run the risk of your chain tool push pin going sideways and becomeing bent/useless. By grinding/filing the wider part of the head off the pin you make it a lot easier to remove because it's being pushed through a hole that is the same diameter as the pin is
After you split the chain, discard the link you hit with the grinder and replace it with a split link or other replacement link designed for the chain