If the harmonica had a plastic comb, you could simply soak the whole thing in lemon juice overnight. Old harmonicas which have pins or rivets typically have pear-wood combs however, and soaking them is not advisable.
Rivets and pins are essentially the same thing. Rivets are hollow-tube pins. A two-part hollow tube would definitely be called a rivet, but you're unlikely to find those on harmonicas.
The procedure for both is similar.
For rivets, use a center punch to guarantee centering a drill, and drill off the end which has the least material until you can pull it through from the other side by using a narrower screw inside, pulled with pliers.
For pins, file off the head on the side with least material and drive it through with a punch, over some table surface with a hole underneath the pin (or a 2x4 with a hole drilled in it, or even a socket).
If you are lucky, the pin/rivet is wide enough to replace with a harmonica bolt and flat nut. Those may not be available though, and rivet pins are easy enough to make anyhow. Tubes can be made by rolling some brass foil. Pins can be made either by flat filing the pointy tip off brass upholstery brads (hold it with pliers) or by balling the end of some copper wire over a gas stove.
Either way, you want a pin/rivet with a head on one side, and about 1mm or 1/16th inch sticking through the other side of the harmonica.
After pressing the pin/rivet through, set the bottom side of the pin on an anvil (any raised piece of metal like a meat tenderizer or side of some pliers). If it's a pin, hammer the other side till it squeezes against the harmonica shell. Next, whether it's a solid pin or hollow tube, tap it in the center with a center punch or large nail. This expands the pin/rivet at the ends and compresses the shell together further. Do this on both sides. It's not difficult so long as you use copper or brass which are soft.
For a clunky but functional hack, even a steel brad will work. In that case don't attempt pining the centers of the ends. Just leave plenty of length so you can twist the other end like a fern frond using needle-nose pliers, then possibly hammer the swirl perpendicular against the harmonica shell.