For instance, I want to measure the angle defined by a carbonyl vector (a vector going through a C=O bond) in my molecule and the z-axis of the simulation box.

Not sure if that tool exists. You could, however, do this:
1) use make_ndx to make an index group of only the two atoms of interest
2) trjconv -f entire.xtc -o two_atoms.gro
3) write a simple script to parse the output .gro trajectory file in order to obtain the angle to the z axis based on cos(theta)=(z2-z1)/r therefore theta=arccos[(z2-z1)/r] where the angle is defined to the negative z direction. If you used constraints for that bond then it will be especially simple, otherwise you will need good old pythagoras to get r. Note1) If you've got lots of angles that you want to average over, the process is the same but with more book-keeping.
Note2) You could also try modifying template.c

Good luck.

PS: if you do write a script, please post it back to the list.

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