Do I need curved RF tracks for a chip antenna?
The datasheet won't address what the transmission line should look like. 

It assumes you use a properly matched transmission line.  

As long as the impedance is correct along the entire length, it doesn't matter if the traces are straight, rounded, curved or zig-zagged.

You may find it difficult to make a transmission line in a particular shape and still be able to guarantee the proper impedance.  That's not something the antenna manufacturer can influence.

The datasheet shows dimensions (or rather, to scale drawings) for the matching network.  You should probably try your best to stick to those recommendations.