Could the acceleration signals be integrated to give X, Y, Z displacement co-ordinates?

It is theoretically possible, after proper co-ordinate transformations and subtraction of the gravitational acceleration, to obtain 3D translation data. In practice, some of the following issues will probably be encountered:

A 'starting point', e.g. 3D reference position, is required from which integration of the 3D acceleration data is started. Similarly, the velocity at the reference position should be either zero or known.

Noise on the acceleration data and small offset errors and/or errors in subtracting gravitational acceleration will also be integrated, and over time will cause increasing (drift) errors in the position and velocity estimate.  In practice, these integration errors will cause the position estimate to become unusable within several seconds.

In conclusion, it depends very much on the (type of) motion to be recorded and the information to be deduced from the position estimate, if this approach is feasible.  Please refer to Xsens' MTi-G for position and orientation tracking.