To get the droll bit out of the way: you know force is a vector from its definition.
To demonstrate that it really is, you would perform experiments: start by attaching three spring scales (like the ones fishermen use to weigh fish) to each other at the same point, and pull the other ends of the scales horizontally at 120 degree angles with equal non-zero force F. The configuration is in the beautiful ascii graphic below, and you can tell the forces are equal by looking at the readings on each scale.
F ----- o
You'll also notice that the point of attachment in the middle stays stationary, that is, the net force is zero.
If F was a scalar, it would be impossible to add or subtract exactly 3 non-zero Fs in whichever order, and get 0 as a result.
Now that you know that force is not a scalar, you would then try to figure out a way to get the three Fs to add up to zero, and you notice that if you pair the direction of each spring to each F, you can get exactly that:
F-----F if you consider the direction each
\ / spring was pulled, you can rearrange
\ / the forces so that they form a loop,
F that is, they add to zero.
You'd then conduct further experiments, in various set-ups, and find that in each case, treating force as a scalar paired with a direction gives the correct result, at which point you would feel justified in saying: for the purposes of calculation, force has both a magnitude and a direction.
A vector, on the other hand, is nothing more than a magnitude paired with a direction, so you have experimentally showed that within the limits of measurement, force is a vector.