Is there an IC with a large number of digitally controllable switches? And if so, how could I control it?
For individual control, twenty-five CD4066 switches would work for you.  They are about $0.50/ea from a distributor.  Or cheaper from China, Futurelec, etc.

If you are controlling this from a microcontroller, it will be more convenient to set up a daisy-chain of thirteen ADG714s, and control them via SPI bus.  These are almost $5 each from distributors, but the switches themselves have better specs.

The cheapskate microcontroller solution is to use the CD4066 switches but control them with digital shift registers like a chain of thirteen 74hc595.