It is theoretically possible to calculate velocity and displacement using initial conditions and acceleration. The equations of motion facilitate this. In practice it is not done this way because of accuracy considerations. See here:
chrobotics accel-position-velocity[
^]
To get good results one needs very accurate accelerometers, a very accurate clock and a very high sampling rate. For small (very small) intervals however it may be possible to get useful values using off the shelf accelerometers. Here is an example:
AN3379 positioning[
^]