Drive 24V Lamps. Will this circuit work?
What you show might be OK, except that there must be flyback diodes in reverse across the relay coils.

No, it's not good enough to just mention this in the text.  You must show such things in the schematic too.  Don't be so lazy.

You should also check whether the transistors can support the full relay coil current.  Apparently your digital outputs will be at 5 V when high.  That puts about 4.3 mA into a transistor base.  Let's say these transistors can be counted on to have a gain of 50 at that operating point (just making up something, your job to check the datasheet), then the maximum supported relay coil current is 215 mA.  Of course it would be good to stay somewhat below that.  Small relays that can switch well beyond the kind of load you show are available with about 60 mA coil current at 5 V, so that all sounds reasonable.

Again, check the transistor and relay datasheets to make sure the above assumptions are valid.