Switching multiple photodiode inputs to an op amp
There's what's called "analog switches and multiplexers", which exactly fulfill the role of your pushbuttons here. Basically, all large silicon chip manufacturers have them (for example,  Texas Instruments).

However, the currents from photodiodes are really small and sensitive. You don't want to switch these, if it can be avoided. These switches can easily "leak" enough current into the current you want to measure (the photocurrent) to void the measurement.

Considering how cheap opamps are (a quad opamp costs insignificantly more than a single opamp in a package) and how little external components are needed (only R1, C1): 

Build an amplifier for each of your diodes, and switch the outputs of these. That way, you'd be switching robust (as we call them, buffered) signals.