Certainly the combustion in the cylinders is the source of the excitation, but I do not agree that the firing order is the primary source of gross engine vibration, which is what most of us will first think of when discussuing engine vibration.
I have owned a number of Triumph twin cylinder motorcycles, all with 360 degree firing intervals (alternate firing). The 1953 T100 is considerably smoother (less vibration) than all others. With the same firing order, the only differences are in the balance factor and the masses of pistons and rods.
Multi cylinder radial engines run with very little vibration, as the collective mass of the rods and pistons moves in something close to a circle and the counterweights can be matched to the collective moment of inertia of the rods and pistons, creating very little secondary vibration.
This is in spite of the fact that the firing order (e.g. 1,3,5,7,2,4,6 in a 7 cylinder engine) would suggest a strong orbital vibration at 1/2 the crankshaft frequency (rpm) if the firing order were to be significant.
Vibration of elements within the engine is a different matter. Certainly, firing order (or rather the distribution of loads along the length of the crankshaft) will influence the magnitude of vibrations within the crankshaft and therefore vibrations within the structure supporting the main bearings.
The suggested example uses two crankshafts, each with a single rod and piston. As such, it is relatively easy to calculate the crankshaft natural frequencies and ensure that they are significantly greater than the firing frequency for each crankshaft.
Similarly, the magnitude of camshaft vibration will be influenced by the sequence of loads (valve opening events) along its length. While it may be tempting to correlate that sequence to firing order, it is dependent on the mechanical layout of the valve actuating mechanism.
In the suggested example with a single cam lobe operating all valve events, all the loads occur at a single location so it should be relatively easy to calculate the camshaft whip frequency and ensure that it is significantly greater than the valve event frequency.
The topic of vibrations of individual components is a very interesting one. I now have CAD software which includes some limited analysis and can calculate natural frequencies of complex parts. Before that, I used excel spreadsheets, calculating the stiffness and mass of each element (e.g. crank journals etc.) to eventually calculate natural frequencies.
While the CAD analysis is much more convenient, there is a certain satisfaction in spending hours on a spreadsheet and then when the part is made, measuring the actual frequency with an accelerometer to correlate with the calculation.