I would like to write some AR mobile app that will show Earth velocity vector, but I have no any experience with astrodynamics. So how to calculate Earth's velocity vector in ECEF, having date and time as input data?
To convert my comment into an answer now that I'm back to my desk: The origin of the ECEF reference frame is the center of the earth, and the X axis goes through the center of the earth and the point on the surface at lat/lon 0/0. Therefore, by definition the earth neither moves nor rotates in the ECEF coordinate space.
To try to provide what you're looking for, I'll offer the JPL Horizons interface. Specify ephemeris type "VECTORS", target body "Sun [Sol] ", coordinate origin set to your current lat/lon location, and a time span starting at the current time with a step size in minutes. That should give you a table of vector positions of the sun relative to your current location. Take the delta between positions (or possibly just use a velocity vector instead of multiple position vectors) and reverse it and you should get your velocity vector relative to the sun.
Obviously this doesn't give you the math to do the calculations yourself, but it should give you the answers and maybe someone else can chime in with the math.