Since the vector itself has a periodicity, then the individual scalar components should also have the same periodicity. So, if we FT one of the scalar components, it's FT should have peaks at the dominant frequencies. Or, for added signal to noise ratio, we can add the squares of the absolute values of the FT's of the components in time.
As an example, here's a vector with a period of 1.3, plus random noise:
Δt = 0.2;
T = 3000.;
list = Table[
Sin[2 π t/1.3 + 2.8] {1, 2, -3, 4} +
RandomReal[{-0.5, 0.5}, 4], {t, Δt, T, Δt}];
Now take the FT of the individual components in time, absolute value them, square them, and add them, and plot:
ftlist = Plus @@ (Abs[Fourier /@ (list\[Transpose])]^2);
ListLinePlot[ftlist, PlotRange -> All]

Now, convert the peak index into period, recovering the 1.3 we originally used (up to quantization error):
T/(Last[Ordering[ftlist[[1 ;; 4000]]]] - 1)
(*1.29983*)
Total /@ ListCorrelate[data, data, 1]
? $\endgroup$FunctionPeriod
might be of help. $\endgroup$