Can the PYNQ package be upgraded locally on the target board?

This will be highly not suggested as the prebuilt image is based on different environment so this is not a good idea to directly upgrade the 2.6 version to 2.7 without rebuilding the image.
It is very easy and only less than 2 hours of install and build to create a custom image of 2.7.
Of cause this is highly depended on the download speed of the Vivado package but once it is downloaded you can build it within 1 hours.

A step by step are posted here.