Yes, and No.
If you use the vendors programming tools the chances are that although they will work in a chain each vendor will only be able to program it's chip.
However there are vendor agnostic 'universal' programming tools, e.g. OpenOCD, which is open source. Commercial ones exist too. Often the work involved in configuring the universal too to work with all the chips is more work then using each vendor tool one at a time.
Most vendor tools allow you to create SVF (Serial Vector Format) files, these files tell third party universal tools and SVF players how to program each of the chips in the chain without the actual tool having to have built in knowledge of how to program the chip.