Can I "integrate" the data from an accelerometer to record a motion trajectory?

No, this won't work in theory or practice because you do not have sensors to capture rotational motion. When you rotate an accelerometer, it is unable to detect that its coordinate system has rotated with respect to the desired coordinate system. What you are trying to do is called inertial navigation. In principle, to do inertial navigation, you need a three-axis accelerometer as well as a three-axis gyro (or angular rate sensor) to capture rotational motion. Then the acceleration data can be converted to displacements in the frame of reference you are using.

In practice, even if you add a gyro, doing this accurately is very difficult because small constant errors in acceleration become very large position errors during the process of integration. The only saving grace in your case is that if you add the assumption that you start and stop in the same place, you may be able to leverage that to calibrate out the drift (again, assuming you add a 3-axis gyro). Although user CortAmmon expressed skepticism that this extra information would be sufficient for calibrating out any drift in the acceleration measurement.

CortAmmon points out that the Northrup Grumman LN200 inertial measurement unit costs US$90,000, and could be expected to have a position error measured in km after the time it takes to do a run. Items like this are not only very expensive, but likely "export controlled" if made in the US. The reason is that Inertial nav units are used in missiles. This gives them the ability to hit a target even when GPS is being jammed.