Schematic help - Unknown capacitors for Raspberry Pi project
C1 is a bypass capacitor for the IC that is just beside. Since not all the parts are labelled, it's difficult to know where it is placed on the board. My guess would be the one on the center right, before the dashed line.

C2 seems to be a bulk cap for the relay, so that there is no power drop when the relay is activated. C7 is also probably a bulk cap for VDD.

Although it's not obvious where it's placed on the board and seems it might not be there at all.

The circuit could work without those caps, although it is usually very recommended to have bypass cap (like C1) as you might start to get into some very difficult to understand issues if they are missing.

For C2 and C7, those are like reservoir of power, the circuit might also work without.

There is something more of a concern on this circuit, is that the relays doesn't have a flyback diode. Unless it's relay with integrated flyback diode, when switched off, the coil of the relay will generate a high voltage spike that might kill the ULN2003 or more components.  Flyback diodes Included in the ULN2003