Is it possible to get the velocity vector for tangential movement using radar?
The answer to your first question is yes if you can resolve the change in bearing or azimuth as well as range in a sequence of measurements, and some variation of a Kalman Filter is used.

The answer to your second question is a lot more interesting because Special Relativity predicts a tangential Doppler shift.  One common scenario used in presenting Special Relativity to students, involves two rocket ships flying at different near light speed velocities comparing their clocks.

This may interest you.

H. W. Thim, "Absence of the relativistic transverse Doppler shift at microwave frequencies," in IEEE Transactions on Instrumentation and Measurement, vol. 52, no. 5, pp. 1660-1664, Oct. 2003.
doi: 10.1109/TIM.2003.817916
Abstract: An experiment is described showing that a 33-GHz microwave signal received by rotating antennas is not exhibiting the frequency shift ("transverse Doppler effect") predicted by the relativistic Doppler formula. The sensitivity of the apparatus used has been tested to be sufficient for detecting frequency shifts as small as 10-3 Hz which corresponds to the value of (v/c)2 = 5.10-14 used in the transverse Doppler shift experiment reported here. From the observed absence of the transverse Doppler shift it is concluded that either the time dilation predicted by the standard theory of special relativity does not exist in reality or, if it does, is a phenomenon which does not depend on relative velocities but may be a function of absolute velocities in the fundamental frame of the isotropic microwave background radiation.
keywords: {Doppler shift;microwave antennas;microwave measurement;33 GHz;absolute velocities;isotropic microwave background radiation;microwave frequencies;relativistic transverse Doppler shift;rotating antennas;time dilation;Doppler effect;Doppler shift;Masers;Microwave antennas;Microwave frequencies;Microwave measurements;Microwave theory and techniques;Receiving antennas;Testing;Transmitting antennas},
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&amp;arnumber=1240186&amp;isnumber=27804